# User:Michiexile/MATH198/Lecture 6

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|                                       | * <math>C</math> has all finite limits.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                         | * <math>C</math> has all finite limits.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       | * <math>C</math> has all finite products and all equalizers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                         | * <math>C</math> has all finite products and all equalizers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
| -                                     | * <math>C</math> has all pullbacks and a terminal object. Also, the following dual statements are equivalent:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                       | * <math>C</math> has all pullbacks and a terminal object.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Also, the following dual statements are equivalent:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       | * <math>C</math> has all finite colimits.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                         | * <math>C</math> has all finite colimits.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       | * <math>C</math> has all finite coproducts and all coequalizers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                         | * <math>C</math> has all finite coproducts and all coequalizers.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
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|                                       | A limit over this diagram is an object <math>C</math> and arrows to all diagram objects. The commutativity conditions for the arrows defined force for us <math>fp_A = p_B = gp_A</math>, and thus, keeping this enforced equation in mind, we can summarize the cone diagram as:                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                         | A limit over this diagram is an object <math>C</math> and arrows to all diagram objects. The commutativity conditions for the arrows defined force for us <math>fp_A = p_B = gp_A</math>, and thus, keeping this enforced equation in mind, we can summarize the cone diagram as:                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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| -                                     | [[Image:EqualizerCone.png]]  Now, the limit condition tells us that this is the least restrictive way we can map into <math>A</math> with some map <math>p</math> such that <math>fp = gp</math>, in that every other way we could map in that way will factor through this way.                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                       | [[Image:EqualizerCone.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Now, the limit condition tells us that this is the least restrictive way we can map into <math>A</math> with some map <math>p</math> such that <math>fp = gp</math>, in that every other way we could map in that way will factor through this way.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
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|                                       | As usual, it is helpful to consider the situation in Set to make sense of any categorical definition: and the situation there is helped by the generalized element viewpoint: the limit object <math>C</math> is one representative of a subobject of <math>A</math> that for the case of Set contains all <math>x\in A: f(x) = g(x)</math>.                                                                                                                                                                                                                                                                                                                                                                                            |                                                         | As usual, it is helpful to consider the situation in Set to make sense of any categorical definition: and the situation there is helped by the generalized element viewpoint: the limit object <math>C</math> is one representative of a subobject of <math>A</math> that for the case of Set contains all <math>x\in A: f(x) = g(x)</math>.                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                       | Hence the word we use for this construction: the limit of the diagram above is the ''equalizer of <math>f, g</math>''. It captures the idea of a maximal subset unable to distinguish two given functions, and it introduces a categorical way to define things by equations we require them to respect.                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                         | Hence the word we use for this construction: the limit of the diagram above is the ''equalizer of <math>f, g</math>''. It captures the idea of a maximal subset unable to distinguish two given functions, and it introduces a categorical way to define things by equations we require them to respect.                                                                                                                                                                                                                                                                                                                                                                                                                                |
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| -                                     | One important special case of the equalizer is the ''kernel'': in a category with a null object, we have a distinguished, unique, member <math>0</math> of any homset given by the compositions of the unique arrows to and from the null object. We define ''the kernel'' <math>Ker(f)</math> of an arrow <math>f</math> to be the equalizer of <math>f, 0</math>. Keeping in mind the arrow-centric view on categories, we tend to denot the arrow from <math>Ker(f)</math> to the source of <math>f</math> by <math>ker(f)</math>.                                                                                                                                                                                                   | +                                                       | One important special case of the equalizer is the ''kernel'': in a category with a null object, we have a distinguished, unique, member <math>0</math> of any homset given by the compositions of the unique arrows to and from the null object. We define ''the kernel'' <math>Ker(f)</math> of an arrow <math>f</math> to be the equalizer of <math>f, 0</math>. Keeping in mind the arrow-centric view on categories, we tend to denote the arrow from <math>Ker(f)</math> to the source of <math>f</math> by <math>ker(f)</math>.                                                                                                                                                                                                  |
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|                                       | In the category of vector spaces, and linear maps, the map <math>0</math> really is the constant map taking the value <math>0</math> everywhere. And the kernel of a linear map <math>f:U\to V</math> is the equalizer of <math>f,0</math>. Thus it is some vector space <math>W</math> with a map <math>i:W\to U</math> such that <math>fi = 0i = 0</math>, and any other map that fulfills this condition factors through <math>W</math>. Certainly the vector space <math>\{u\in U: f(u)=0\}</math> fulfills the requisite condition, nothing larger will do, since then the map composition wouldn't be 0, and nothing smaller will do, since then the maps factoring this space through the smaller candidate would not be unique. |                                                         | In the category of vector spaces, and linear maps, the map <math>0</math> really is the constant map taking the value <math>0</math> everywhere. And the kernel of a linear map <math>f:U\to V</math> is the equalizer of <math>f,0</math>. Thus it is some vector space <math>W</math> with a map <math>i:W\to U</math> such that <math>fi = 0i = 0</math>, and any other map that fulfills this condition factors through <math>W</math>. Certainly the vector space <math>\{u\in U: f(u)=0\}</math> fulfills the requisite condition, nothing larger will do, since then the map composition wouldn't be 0, and nothing smaller will do, since then the maps factoring this space through the smaller candidate would not be unique. |
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|                                       | A coequalizer                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                         | A coequalizer                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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|                                       | [[Image:CoequalizerCoCone.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                         | [[Image:CoequalizerCoCone.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
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|                                       | has to fulfill that <math>i_Bf = i_A = i_Bg</math>. Thus, writing <math>q=i_B</math>, we get an object with an arrow (actually, an epimorphism out of <math>B</math>) that identifies <math>f</math> and <math>g</math>. Hence, we can think of <math>i_B:B\to Q</math> as catching the notion of inducing equivalence classes from the functions.                                                                                                                                                                                                                                                                                                                                                                                      |                                                         | has to fulfill that <math>i_Bf = i_A = i_Bg</math>. Thus, writing <math>q=i_B</math>, we get an object with an arrow (actually, an epimorphism out of <math>B</math>) that identifies <math>f</math> and <math>g</math>. Hence, we can think of <math>i_B:B\to Q</math> as catching the notion of inducing equivalence classes from the functions.                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                       | [[Image:PreimageDiagram.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                         | [[Image:PreimageDiagram.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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| -                                     | where <math>i</math> is a monomorphism representing the subobject, we need to find an object <math>V</math> with a monomorphism injecting it into <math>U</math> such that the map <math>fi: U\to S</math> factors through <math>T</math>. Thus we're looking for dotted maps making the diagram commute, in a universal manner.                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                       | where <math>i</math> is a monomorphism representing the subobject, we need to find an object <math>V</math> with a monomorphism injecting it into <math>U</math> such that the map <math>\bar fj: V\to S</math> factors through <math>T</math>. Thus we're looking for dotted maps making the diagram commute, in a universal manner.                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                       | The maximality of the subobject means that any other subobject of <math>U</math> that can be factored through <math>T</math> should factor through <math>V</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                         | The maximality of the subobject means that any other subobject of <math>U</math> that can be factored through <math>T</math> should factor through <math>V</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                       | [[Image:PullbackDiagram.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                         | [[Image:PullbackDiagram.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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| -                                     | By the definition of a limit, this means that the pullback is an object <math>D</math> with maps <math>\bar f: D\to B</math>, <math>\bar g: D\to A</math> and <math>f\bar g = g\bar f : D\to C</math>, such that any other such object factors through this.                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                       | By the definition of a limit, this means that the pullback is an object <math>P</math> with maps <math>\bar f: P\to B</math>, <math>\bar g: P\to A</math> and <math>f\bar g = g\bar f : P\to C</math>, such that any other such object factors through this.                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
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|                                       | For the diagram <math>U\rightarrow^f S \leftarrow^i T</math>, with <math>i:T\to S</math> one representative monomorphism for the subobject, we get precisely the definition above for the inverse image.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                         | For the diagram <math>U\rightarrow^f S \leftarrow^i T</math>, with <math>i:T\to S</math> one representative monomorphism for the subobject, we get precisely the definition above for the inverse image.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
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|                                       | ===Free and forgetful functors===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                         | ===Free and forgetful functors===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Recall how we defined a free monoid as all strings of some alphabet, with concatenation of strings the monoidal operation. And recall how we defined the free category on a graph as the category of paths in the graph, with path concatenation as the operation.                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | The reason we chose the word ''free'' to denote both these cases is far from a coincidence: by this point nobody will be surprised to hear that we can unify the idea of generating the most general object of a particular algebraic structure into a single categorical idea.                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | The idea of the free constructions, classically, is to introduce as few additional relations as possible, while still generating a valid object of the appropriate type, given a set of generators we view as placeholders, as symbols. Having a minimal amount of relations allows us to introduce further relations later, by imposing new equalities by mapping with surjections to other structures.                                                                                                                                                                                                                                                                                                                                |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | One of the first observations in each of the cases we can do is that such a map ends up being completely determined by where the generators go - the symbols we use to generate. And since the free structure is made to fulfill the axioms of whatever structure we're working with, these generators combine, even after mapping to some other structure, in a way compatible with all structure.                                                                                                                                                                                                                                                                                                                                     |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | To make solid categorical sense of this, however, we need to couple the construction of a free algebraic structure from a set (or a graph, or...) with another construction: we can define the ''forgetful functor'' from monoids to sets by just picking out the elements of the monoid as a set; and from categories to graph by just picking the underlying graph, and forgetting about the compositions of arrows.                                                                                                                                                                                                                                                                                                                  |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Now we have what we need to pinpoint just what kind of a functor the ''free widget generated by''-construction does. It's a functor <math>F: C\to D</math>, coupled with a forgetful functor <math>U: D\to C</math> such that any map <math>S\to U(N)</math> in <math>C</math> induces one unique mapping <math>F(S)\to N</math> in <math>D</math>.                                                                                                                                                                                                                                                                                                                                                                                     |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | For the case of monoids and sets, this means that if we take our generating set, and map it into the set of elements of another monoid, this generates a unique mapping of the corresponding monoids.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | This is all captured by a similar kind of diagrams and uniquely existing maps argument as the previous object or morphism properties were defined with. We'll show the definition for the example of monoids.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | '''Definition''' A ''free'' monoid on a generating set <math>X</math> is a monoid <math>F(X)</math> such that there is an inclusion <math>i_X: X\to UF(X)</math> and for every function <math>f : X\to U(M)</math> for some other monoid <math>M</math>, there is a unique homomorphism <math>g:F(X)\to M</math> such that <math>f=U(g) i_X</math>, or in other words such that this diagram commutes:                                                                                                                                                                                                                                                                                                                                  |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:FreeMonoid.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
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|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | We can construct a map <math>\phi:Hom_{Mon}(F(X),M) \to Hom_{Set}(X,U(M))</math> by <math>\phi: g\mapsto U(g)\circ i_X</math>. The above definition says that this map is an isomorphism.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
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|                                       | ===Adjunctions===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                         | ===Adjunctions===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
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| -                                     | * Free and forgetful                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                       | Modeling on the way we construct free and forgetful functors, we can form a powerful categorical concept, which ends up generalizing much of what we've already seen - and also leads us on towards monads.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
| -                                     | * Curry and uncurry                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | We draw on the definition above of free monoids to give a preliminary definition. This will be replaced later by an equivalent definition that gives more insight.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | '''Definition''' A pair of functors,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointPair.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | is called an ''adjoint pair'' or an ''adjunction'', with <math>F</math> called the ''left adjoint'' and <math>U</math> called the ''right adjoint'' if there is natural transformation <math>\eta: 1\to UF</math>, and for every <math>f:A\to U(B)</math>, there is a unique <math>g: F(A)\to B</math> such that the diagram below commutes.                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointDefinition.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | The natural transformation <math>\eta</math> is called the ''unit'' of the adjunction.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | This definition, however, has a significant amount of asymmetry: we can start with some <math>f:A\to U(B)</math> and generate a <math>g: F(A)\to B</math>, while there are no immediate guarantees for the other direction. However, there is a proposition we can prove leading us to a more symmetric statement:                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | '''Proposition''' For categories and functors                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointPair.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | the following conditions are equivalent:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # <math>F</math> is left adjoint to <math>U</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # For any <math>c\in C_0</math>, <math>d\in D_0</math>, there is an isomorphism <math>\phi: Hom_D(Fc, d) \to Hom_C(c,Ud)</math>, natural in both <math>c</math> and <math>d</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | moreover, the two conditions are related by the formulas                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | * <math>\phi(g) = U(g) \circ \eta_c</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | * <math>\eta_c = \phi(1_{Fc})</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | '''Proof sketch'''                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | For (1 implies 2), the isomorphism is given by the end of the statement, and it is an isomorphism exactly because of the unit property - viz. that every <math>f:A\to U(B)</math> generates a unique <math>g: F(A)\to B</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Naturality follows by building the naturality diagrams                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointNaturalFirst.png]] [[Image:AdjointNaturalSecond.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | and chasing through with a <math>f: Fc\to d</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | For (2 implies 1), we start out with a natural isomorphism <math>\phi</math>. We find the necessary natural transformation <math>\eta_c</math> by considering <math>\phi: Hom(Fc,Fc) \to Hom(c, UFc)</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | QED.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | By dualizing the proof, we get the following statement:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | '''Proposition''' For categories and functors                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointPair.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | the following conditions are equivalent:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # For any <math>c\in C_0</math>, <math>d\in D_0</math>, there is an isomorphism <math>\phi: Hom_D(Fc, d) \to Hom_C(c,Ud)</math>, natural in both <math>c</math> and <math>d</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # There is a natural transformation <math>\epsilon: FU \to 1_D</math> with the property that for any <math>g: F(c) \to d</math> there is a unique <math>f: c\to U(d)</math> such that <math>g = \epsilon_D\circ F(f)</math>, as in the diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointDualDefinition.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | moreover, the two conditions are related by the formulas                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | * <math>\psi(f) = \epsilon_D\circ F(f)</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | * <math>\epsilon_d = \psi(1_{Ud})</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | where <math>\psi = \phi^{-1}</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Hence, we have an equivalent definition with higher generality, more symmetry and more ''horsepower'', as it were:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | '''Definition''' An ''adjunction'' consists of functors                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointPair.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | and a natural isomorphism                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | [[Image:AdjointIso.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | The ''unit'' <math>\eta</math> and the ''counit'' <math>\epsilon</math> of the adjunction are natural transformations given by:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | * <math>\eta: 1_C\to UF: \eta_c = \phi(1_{Fc})</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | * <math>\epsilon: FU\to 1_D: \epsilon_d = \psi(1_{Ud})</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | ----                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Some of the examples we have had difficulties fitting into the limits framework show up as adjunctions:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | The ''free'' and ''forgetful'' functors are adjoints; and indeed, a more natural definition of what it means to be free is that it is a left adjoint to some forgetful functor.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Curry and uncurry, in the definition of an exponential object are an adjoint pair. The functor <math>-\times A: X\mapsto X\times A</math> has right adjoint <math>-^A: Y\mapsto Y^A</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | ====Notational aid====                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | One way to write the adjoint is as a ''bidirectional rewrite rule'':                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | <math>\frac{F(X) \to Y}{X\to G(Y)}</math>,                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | where the statement is that the hom sets indicated by the upper and lower arrow, respectively, are transformed into each other by the unit and counit respectively. The left adjoint is the one that has the functor application on the left hand side of this diagram, and the right adjoint is the one with the functor application to the right.                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| Line 110:                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | Line 221:                                               |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       | ===Homework===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                         | ===Homework===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | Complete homework is 6 out of 11 exercises.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                       | # Prove that an equalizer is a monomorphism.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                         | # Prove that an equalizer is a monomorphism.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       | # Prove that a coequalizer is an epimorphism.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                         | # Prove that a coequalizer is an epimorphism.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| -                                     | # Prove that given any relation <math>R\subseteq X\times X</math>, its completion to an equivalence relation is the kernel of the coequalizer of the component maps of the relation                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     | +                                                       | # Prove that given any relation <math>R\subseteq X\times X</math>, its completion to an equivalence relation is the kernel of the coequalizer of the component maps of the relation. For the purpose of this, we define kernels in Set as the equalizer of <math>f\circ p_1, f\circ p_2: A\times A\to B</math>. (A more general definition of a kernel, independent of zero objects, can be found using a definition of quotients by equivalence relations, and having the kernel be a universal object to factor the quotient through).                                                                                                                                                                                                |
|                                       | # Prove that if the right arrow in a pullback square is a mono, then so is the left arrow. Thus the intersection as a pullback really is a subobject.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                         | # Prove that if the right arrow in a pullback square is a mono, then so is the left arrow. Thus the intersection as a pullback really is a subobject.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       | # Prove that if both the arrows in the pullback 'corner' are mono, then the arrows of the pullback cone are all mono.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                         | # Prove that if both the arrows in the pullback 'corner' are mono, then the arrows of the pullback cone are all mono.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       | # What is the pullback in the category of posets?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                         | # What is the pullback in the category of posets?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                       | # What is the pushout in the category of posets?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                         | # What is the pushout in the category of posets?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # Prove that the exponential and the product functors above are adjoints. What are the unit and counit?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # (worth 4pt) Consider the unique functor <math>!:C\to 1</math> to the terminal category.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | ## Does it have a left adjoint? What is it?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | ## Does it have a right adjoint? What is it?                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # * Prove the propositions in the text.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | # (worth 4pt) Suppose                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | :[[Image:AdjointPair.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | :is an adjoint pair. Find a natural transformation <math>FUF\to F</math>. Conclude that there is a natural transformation <math>\mu: UFUF\to UF</math>. Prove that this is associative, in other words that the diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | :[[Image:AdjointMuAssociative.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | :commutes. Prove that the unit of the adjunction forms a unit for this <math>\mu</math>, in other words, that the diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | :[[Image:AdjointMuUnit.png]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                       |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                       | :commutes.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |

## Current revision

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## Contents

### 1 Useful limits and colimits

With the tools of limits and colimits at hand, we can start using these to introduce more category theoretical constructions - and some of these turn out to correspond to things we've seen in other areas.

Possibly among the most important are the equalizers and coequalizers (with kernel (nullspace) and images as special cases), and the pullbacks and pushouts (with which we can make explicit the idea of inverse images of functions).

One useful theorem to know about is:

Theorem The following are equivalent for a category C:

• C has all finite limits.
• C has all finite products and all equalizers.
• C has all pullbacks and a terminal object.

Also, the following dual statements are equivalent:

• C has all finite colimits.
• C has all finite coproducts and all coequalizers.
• C has all pushouts and an initial object.

For this theorem, we can replace finite with any other cardinality in every place it occurs, and we will still get a valid theorem.

====Equalizer, coequalizer==== Consider the equalizer diagram:

A limit over this diagram is an object C and arrows to all diagram objects. The commutativity conditions for the arrows defined force for us fpA = pB = gpA, and thus, keeping this enforced equation in mind, we can summarize the cone diagram as:

Now, the limit condition tells us that this is the least restrictive way we can map into A with some map p such that fp = gp, in that every other way we could map in that way will factor through this way.

As usual, it is helpful to consider the situation in Set to make sense of any categorical definition: and the situation there is helped by the generalized element viewpoint: the limit object C is one representative of a subobject of A that for the case of Set contains all $x\in A: f(x) = g(x)$.

Hence the word we use for this construction: the limit of the diagram above is the equalizer of f,g. It captures the idea of a maximal subset unable to distinguish two given functions, and it introduces a categorical way to define things by equations we require them to respect.

One important special case of the equalizer is the kernel: in a category with a null object, we have a distinguished, unique, member 0 of any homset given by the compositions of the unique arrows to and from the null object. We define the kernel Ker(f) of an arrow f to be the equalizer of f,0. Keeping in mind the arrow-centric view on categories, we tend to denote the arrow from Ker(f) to the source of f by ker(f).

In the category of vector spaces, and linear maps, the map 0 really is the constant map taking the value 0 everywhere. And the kernel of a linear map $f:U\to V$ is the equalizer of f,0. Thus it is some vector space W with a map $i:W\to U$ such that fi = 0i = 0, and any other map that fulfills this condition factors through W. Certainly the vector space $\{u\in U: f(u)=0\}$ fulfills the requisite condition, nothing larger will do, since then the map composition wouldn't be 0, and nothing smaller will do, since then the maps factoring this space through the smaller candidate would not be unique.

Hence, $Ker(f) = \{u\in U: f(u) = 0\}$ just like we might expect.

Dually, we get the coequalizer as the colimit of the equalizer diagram.

A coequalizer

has to fulfill that iBf = iA = iBg. Thus, writing q = iB, we get an object with an arrow (actually, an epimorphism out of B) that identifies f and g. Hence, we can think of $i_B:B\to Q$ as catching the notion of inducing equivalence classes from the functions.

This becomes clear if we pick out one specific example: let $R\subseteq X\times X$ be an equivalence relation, and consider the diagram

where r1 and r2 are given by the projection of the inclusion of the relation into the product onto either factor. Then, the coequalizer of this setup is an object X / R such that whenever x˜Ry, then q(x) = q(y).

#### 1.1 Pullbacks

The preimage f − 1(T) of a subset $T\subseteq S$ along a function $f:U\to S$ is a maximal subset $V\subseteq U$ such that for every $v\in V: f(v)\in T$.

We recall that subsets are given by (equivalence classes of) monics, and thus we end up being able to frame this in purely categorical terms. Given a diagram like this:

where i is a monomorphism representing the subobject, we need to find an object V with a monomorphism injecting it into U such that the map $\bar fj: V\to S$ factors through T. Thus we're looking for dotted maps making the diagram commute, in a universal manner.

The maximality of the subobject means that any other subobject of U that can be factored through T should factor through V.

Suppose U,V are subsets of some set W. Their intersection $U\cap V$ is a subset of U, a subset of V and a subset of W, maximal with this property.

Translating into categorical language, we can pick representatives for all subobjects in the definition, we get a diagram with all monomorphisms:

where we need the inclusion of $U\cap V$ into W over U is the same as the inclusion over V.

Definition A pullback of two maps $A \rightarrow^f C \leftarrow^g B$ is the limit of these two maps, thus:

By the definition of a limit, this means that the pullback is an object P with maps $\bar f: P\to B$, $\bar g: P\to A$ and $f\bar g = g\bar f : P\to C$, such that any other such object factors through this.

For the diagram $U\rightarrow^f S \leftarrow^i T$, with $i:T\to S$ one representative monomorphism for the subobject, we get precisely the definition above for the inverse image.

For the diagram $U\rightarrow W \leftarrow V$ with both map monomorphisms representing their subobjects, the pullback is the intersection.

#### 1.2 Pushouts

Often, especially in geometry and algebra, we construct new structures by gluing together old structures along substructures. Possibly the most popularly known example is the Möbius band: we take a strip of paper, twist it once and glue the ends together.

Similarily, in algebraic contexts, we can form amalgamated products that do roughly the same.

All these are instances of the dual to the pullback:

Definition A pushout of two maps $A\leftarrow^f C\rightarrow^g B$ is the co-limit of these two maps, thus:

Hence, the pushout is an object D such that C maps to the same place both ways, and so that, contingent on this, it behaves much like a coproduct.

### 2 Free and forgetful functors

Recall how we defined a free monoid as all strings of some alphabet, with concatenation of strings the monoidal operation. And recall how we defined the free category on a graph as the category of paths in the graph, with path concatenation as the operation.

The reason we chose the word free to denote both these cases is far from a coincidence: by this point nobody will be surprised to hear that we can unify the idea of generating the most general object of a particular algebraic structure into a single categorical idea.

The idea of the free constructions, classically, is to introduce as few additional relations as possible, while still generating a valid object of the appropriate type, given a set of generators we view as placeholders, as symbols. Having a minimal amount of relations allows us to introduce further relations later, by imposing new equalities by mapping with surjections to other structures.

One of the first observations in each of the cases we can do is that such a map ends up being completely determined by where the generators go - the symbols we use to generate. And since the free structure is made to fulfill the axioms of whatever structure we're working with, these generators combine, even after mapping to some other structure, in a way compatible with all structure.

To make solid categorical sense of this, however, we need to couple the construction of a free algebraic structure from a set (or a graph, or...) with another construction: we can define the forgetful functor from monoids to sets by just picking out the elements of the monoid as a set; and from categories to graph by just picking the underlying graph, and forgetting about the compositions of arrows.

Now we have what we need to pinpoint just what kind of a functor the free widget generated by-construction does. It's a functor $F: C\to D$, coupled with a forgetful functor $U: D\to C$ such that any map $S\to U(N)$ in C induces one unique mapping $F(S)\to N$ in D.

For the case of monoids and sets, this means that if we take our generating set, and map it into the set of elements of another monoid, this generates a unique mapping of the corresponding monoids.

This is all captured by a similar kind of diagrams and uniquely existing maps argument as the previous object or morphism properties were defined with. We'll show the definition for the example of monoids.

Definition A free monoid on a generating set X is a monoid F(X) such that there is an inclusion $i_X: X\to UF(X)$ and for every function $f : X\to U(M)$ for some other monoid M, there is a unique homomorphism $g:F(X)\to M$ such that f = U(g)iX, or in other words such that this diagram commutes:

We can construct a map $\phi:Hom_{Mon}(F(X),M) \to Hom_{Set}(X,U(M))$ by $\phi: g\mapsto U(g)\circ i_X$. The above definition says that this map is an isomorphism.

### 3 Adjunctions

Modeling on the way we construct free and forgetful functors, we can form a powerful categorical concept, which ends up generalizing much of what we've already seen - and also leads us on towards monads.

We draw on the definition above of free monoids to give a preliminary definition. This will be replaced later by an equivalent definition that gives more insight.

Definition A pair of functors,

is called an adjoint pair or an adjunction, with F called the left adjoint and U called the right adjoint if there is natural transformation $\eta: 1\to UF$, and for every $f:A\to U(B)$, there is a unique $g: F(A)\to B$ such that the diagram below commutes.

The natural transformation η is called the unit of the adjunction.

This definition, however, has a significant amount of asymmetry: we can start with some $f:A\to U(B)$ and generate a $g: F(A)\to B$, while there are no immediate guarantees for the other direction. However, there is a proposition we can prove leading us to a more symmetric statement:

Proposition For categories and functors

the following conditions are equivalent:

1. F is left adjoint to U.
2. For any $c\in C_0$, $d\in D_0$, there is an isomorphism $\phi: Hom_D(Fc, d) \to Hom_C(c,Ud)$, natural in both c and d.

moreover, the two conditions are related by the formulas

• $\phi(g) = U(g) \circ \eta_c$
• ηc = φ(1Fc)

Proof sketch For (1 implies 2), the isomorphism is given by the end of the statement, and it is an isomorphism exactly because of the unit property - viz. that every $f:A\to U(B)$ generates a unique $g: F(A)\to B$.

Naturality follows by building the naturality diagrams

and chasing through with a $f: Fc\to d$.

For (2 implies 1), we start out with a natural isomorphism φ. We find the necessary natural transformation ηc by considering $\phi: Hom(Fc,Fc) \to Hom(c, UFc)$.

QED.

By dualizing the proof, we get the following statement:

Proposition For categories and functors

the following conditions are equivalent:

1. For any $c\in C_0$, $d\in D_0$, there is an isomorphism $\phi: Hom_D(Fc, d) \to Hom_C(c,Ud)$, natural in both c and d.
2. There is a natural transformation $\epsilon: FU \to 1_D$ with the property that for any $g: F(c) \to d$ there is a unique $f: c\to U(d)$ such that $g = \epsilon_D\circ F(f)$, as in the diagram

moreover, the two conditions are related by the formulas

• $\psi(f) = \epsilon_D\circ F(f)$
• εd = ψ(1Ud)

where ψ = φ − 1.

Hence, we have an equivalent definition with higher generality, more symmetry and more horsepower, as it were:

Definition An adjunction consists of functors

and a natural isomorphism

The unit η and the counit ε of the adjunction are natural transformations given by:

• $\eta: 1_C\to UF: \eta_c = \phi(1_{Fc})$
• $\epsilon: FU\to 1_D: \epsilon_d = \psi(1_{Ud})$.

Some of the examples we have had difficulties fitting into the limits framework show up as adjunctions:

The free and forgetful functors are adjoints; and indeed, a more natural definition of what it means to be free is that it is a left adjoint to some forgetful functor.

Curry and uncurry, in the definition of an exponential object are an adjoint pair. The functor $-\times A: X\mapsto X\times A$ has right adjoint $-^A: Y\mapsto Y^A$.

#### 3.1 Notational aid

One way to write the adjoint is as a bidirectional rewrite rule:

$\frac{F(X) \to Y}{X\to G(Y)}$,

where the statement is that the hom sets indicated by the upper and lower arrow, respectively, are transformed into each other by the unit and counit respectively. The left adjoint is the one that has the functor application on the left hand side of this diagram, and the right adjoint is the one with the functor application to the right.

### 4 Homework

Complete homework is 6 out of 11 exercises.

1. Prove that an equalizer is a monomorphism.
2. Prove that a coequalizer is an epimorphism.
3. Prove that given any relation $R\subseteq X\times X$, its completion to an equivalence relation is the kernel of the coequalizer of the component maps of the relation. For the purpose of this, we define kernels in Set as the equalizer of $f\circ p_1, f\circ p_2: A\times A\to B$. (A more general definition of a kernel, independent of zero objects, can be found using a definition of quotients by equivalence relations, and having the kernel be a universal object to factor the quotient through).
4. Prove that if the right arrow in a pullback square is a mono, then so is the left arrow. Thus the intersection as a pullback really is a subobject.
5. Prove that if both the arrows in the pullback 'corner' are mono, then the arrows of the pullback cone are all mono.
6. What is the pullback in the category of posets?
7. What is the pushout in the category of posets?
8. Prove that the exponential and the product functors above are adjoints. What are the unit and counit?
9. (worth 4pt) Consider the unique functor $!:C\to 1$ to the terminal category.
1. Does it have a left adjoint? What is it?
2. Does it have a right adjoint? What is it?
10. * Prove the propositions in the text.
11. (worth 4pt) Suppose
is an adjoint pair. Find a natural transformation $FUF\to F$. Conclude that there is a natural transformation $\mu: UFUF\to UF$. Prove that this is associative, in other words that the diagram
commutes. Prove that the unit of the adjunction forms a unit for this μ, in other words, that the diagram
commutes.
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