(Warning: I’m trying to talk about things I don’t really understand in this post, so feel free to correct me if you see a statement that’s obviously wrong.)
Why are continuous functions the “correct” notion of homomorphism between topological spaces?
The “obvious” way to define homomorphisms for a large class of objects involves thinking of them as “sets with extra structure,” and then homomorphisms are functions that preserve that extra structure. In category theory this is formalized by the notion of a concrete category, i.e. a category with a good notion of forgetful functor. For topological spaces this is the functor which gives the set of points.
However, a naive interpretation of “preserving additional structure” suggests that homomorphisms between topological spaces should be open maps, and this isn’t the case. So what gives?
I’m not sure. But rather than take the concrete perspective I’d like to talk about another general principle for defining a good notion of homomorphism.
A functor from a category to another category sends objects and morphisms in the former to objects and morphisms in the latter in a way that preserves identity morphisms and composition. To a first approximation, a functorial construction is one that is “canonical.” Examples in large categories include
- The construction of the dual space of a vector space , which is contravariant,
- The construction of the free group generated by a set of symbols,
- The construction of the fundamental group of a pointed space,
- Hom functors; subsuming the first example, since where is the base field, the dual space construction is such a functor, and Harrison gives another example: the set of all colorings of a graph with colors is ,
- Any combinatorial species; for example, the construction of the set of all total orders on a set.
In all of these examples we already had a good notion of homomorphism between the objects we cared about, and we used that as the morphisms of our category. But one of the great things about category theory is how the same concepts reappear at different “levels.”
Objects as small categories
Although the Unapologetic Mathematician did it, it’s somewhat unusual to define categories before defining groups, but if you can be convinced that categories are useful without seeing any abstract algebra then you might like this definition.
Definition: A group is a small category with one object and all morphisms invertible.
One automatically obtains the definition of a group homomorphism from the definition of a functor, since the identity and composition of morphisms is preserved. The really nice thing about this point of view, as John Armstrong makes clear, is that all sorts of other ideas in group theory can be discussed very succinctly using categorical language. A group action of a group is just a functor from to the category of sets and functions; a representation of is a functor to the category of vector spaces and linear transformations.
It is possible, but less trivial, to define rings and algebras this way as well; again one automatically gets the definition of homomorphism for free. The definition easily extends, however, to a homomorphism of group actions, since any group action of a group on a set can be viewed as a category with object set and morphism set , where a morphism is the arrow from to . Group actions define equivalence relations on , and in fact equivalence relations are also categories.
Definition: An equivalence relation on a set is a small category with object set and at most one morphism between objects such that every morphism is invertible.
What is nice about this perspective is that all of the equivalence relation axioms fall very naturally out of the category axioms. Composition is equivalent to transitivity, the existence of identity morphisms is equivalent to reflexivity, and symmetry is equivalent to invertibility. Functors between categories of the above form are morphisms of equivalence relations (although people don’t seem to talk about those much), which are precisely the functions that preserve equivalence. A common generalization of the above definitions is the following.
Definition: A groupoid is a small category with all morphisms invertible.
Thus a group is a one-object groupoid. Groupoids are a natural way to study objects whose “symmetries” only compose under certain conditions; for example, the set of permissible moves in the 15 puzzle form a groupoid. Weinstein has a nice introduction to groupoids here.
Groupoids are very cool; they have an intriguing notion of cardinality based on symmetry considerations that extends to some categories. There’s a lot of mysterious stuff going on here; see, for example, John Baez’s summary.
Anyway, back to stuff I sort of understand. Equivalence relations are special cases of relations; let’s talk about another important type of relation.
Definition: A partially ordered set is a small category with at most one morphism between distinct objects such that no morphisms besides the identity are invertible.
Again, the usual axioms for a poset fall out of the category axioms: Composition is equivalent to transitivity, the existence of identity morphisms is equivalent to reflexivity, and antisymmetry is equivalent to non-invertibility. Again, functors between such categories define homomorphisms of posets, which are precisely the order-preserving or monotone / monotonic functions.
Many of the important concepts of poset theory correspond to ideas first understood (at least if I’ve got my history right) for large categories; for example, infimums and supremums are none other than coproducts and products, and initial and terminal objects are none other than greatest and smallest elements. A lattice is a poset with all finite products and coproducts (although in lattice theory they’re called joins and meets as a generalization of unions and intersections in Boolean lattices).
Topological spaces are usually defined in terms of their lattice of open sets, which are characterized by the fact that they are a sublattice of the Boolean lattice of some set having finite meets, arbitrary joins, a smallest element (the empty set), and a greatest element (the whole set). This perspective suggests the direction of pointless topology, which has interesting things to say regarding why the definition of a continuous function seems “backwards.” Here’s what the Wikipedia article seems to be saying.
Definition: A frame is a lattice such that finite meets distribute over arbitrary joins (i.e. finite coproducts distribute over arbitrary products).
Hence any lattice of open sets is a frame, although the converse isn’t true. Since coproducts and products can be defined by means of diagrams, I think there is some general abstract nonsense that justifies looking at functors (morphisms) between frames that preserve finite meets and arbitrary joins, although I don’t have the category theory background to know if this is really true or not. Anyway, this defines a category of frames with a “natural” notion of homomorphism.
Given two topological spaces in the category of topological spaces and continuous functions, there is a contravariant functor taking to its lattice of open sets and taking a continuous function to the function defined by taking preimages. The contravariance of this functor is the essence of the “backwardness” of the definition of continuous function.
So where do contravariant functors come from? A natural place to look for them is in Hom functors of the form for some fixed object . A moment’s thought suggests setting with a topology generated by the open set . Continuous functions are in one-to-one correspondence with open sets in ; just take the preimage of . It follows that is essentially the contravariant functor . (There’s an enrichment here, but I’m not sure what the details are.)
So this seems to be more or less what’s going on:
- Given a set in the category of sets and functions, where gives a functor from to the power set of , i.e. the Boolean lattice . To recover the order structure one should move to the category of posets and require that ; this induces the usual order on . (Again, there's an enrichment here, but I’m not sure what the details are.)
- If you want to pick out certain kinds of subsets over others, you need to control the preimage of in the above construction.
- If you want your Boolean lattices to have extra properties (finite meets, arbitrary joins) and morphisms preserving those properties, this restricts the type of functions that can be lifted by the functor above in a way naturally involving preimages.
Or something like that. If someone could let me know if I’m on the right track and / or correct something obviously wrong that I’ve said, that would be great. Alternately, propose an entirely different definition of a topological space as a small category; I think an approach is possible through the Kuratowski closure axioms. I’d also love it if someone could tell me what the Yoneda lemma has to say about all this.