# Tagged Questions

Various structures are studied in category theory using properties of objects and morphisms between them. Many construction are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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### What's more robust than a structural homomorphisms?

This question isn't category theory; but, category theoreticians tend to be interested in mathematical structure, so I thought the answer might exist within that knowledge base. Given two ...
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108 views

### Is “cofunctor” an accepted term for contravariant functors?

People are used to the prefix co- flipping arrows in a concept1, and I have seen people using cofunctor to mean a functor that flips arrows, i.e. that takes $A \to B$ to $FB \to FA$. I know this ...
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### What does a proof in an internal logic actually look like?

The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. ...
1answer
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### Borceux. Handbook of Categorical Algebra I. Proposition 3.4.2.

I'm trying to understand proposition 3.' HoCA (vol. I). Proposition 3.4.2 Consider a functor $F\colon \mathcal A \to \mathcal B$ with both a left adjoint functor $G$ and a right adjoint functor $H$. ...
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### Do these definitions of congruences on categories have the same result in this context?

Let $\mathcal{D}$ be a small category and let $A=A\left(\mathcal{D}\right)$ be its set of arrows. Define $P$ on $A$ by: $fPg\Leftrightarrow\left[f\text{ and }g\text{ are parallel}\right]$ and let ...
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### When is $T$-Alg monoidal closed?

Given a category $\mathcal{V}$ and a monad $(T,\eta,\mu)$, what would be the sufficient conditions on $\mathcal{V}$ and $T$, for the category of $T$ algebras to be monoidal closed? (I'm pretty sure ...
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97 views

### Functoriality of the Fundamental group

The fundamental group is a functor from the category of pointed topological spaces to the category of groups. Therefore every base-point preserving continuous function $f$ between pointed ...
2answers
74 views

### Do we ever study “mixed” categories?

Consider a category whose object class includes the class of all topological spaces and the class of all topological groups. Furthermore, let the hom-sets between any two objects be the usual hom-sets ...
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### Category of adjunctions inducing a particular monad

Every pair $F \dashv G$ of adjoint functors $F: \mathcal C \to \mathcal D$, $G: \mathcal D \to \mathcal C$ induces a monad $\mathbb T = (T,\eta,\mu)$ on $\mathcal C$. Given a monad \$\mathbb T = ...
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### Representable functors preserve limits. [duplicate]

Theorem : Representable functors preserve limits. I'm struggling to see why this is true. It's not obvious to me where I should be actually using the fact that functor in question is representable. ...
2answers
168 views

### how many empty sets are there?

Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"? In other words, there are LOTS of initial objects in the ...
1answer
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### Dagger category generated by $\mathsf{Set}$ viewed as a subcategory of $\mathsf{Rel}$.

Whenever a category $\mathcal{C}$ is being viewed a subcategory of a dagger category $\mathcal{D}$, define that the dagger category generated by $\mathcal{C}$ is the least subcategory of $\mathcal{D}$ ...
0answers
58 views

### Does every smooth surjective function have a smooth right inverse?

If you feel this question might be too broad, let me know and I’ll try to get more specific. If $r \colon I → J$ is a smooth surjective function between perfect subspaces $I$ and $J$ of $ℝ$, can ...
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47 views

### Commuting square of functors

Let $\mathcal{E}$ be a complete and cocomplete category. Given a functor $i: \mathcal{C} \to \mathcal{D}$ between small categories, there is a triple of adjoint functors between their respective ...
1answer
31 views

### Whether the limit on representable functors be non-representable?

I'm looking for examples of the following situation: Let $A$ be a complete and cocomplete category, $B$ is a small category and $T\colon B\to\mathbf{Set}^{A^{op}}$ be a functor, such that for any ...
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66 views

### Group actions and natural isomorphisms

Let $G$ be a group-as-category, and let $S$ be the image of $G$ by the $Hom(G,-)$ functor. The $Hom$ functor defines a bijection $\chi: G \to S$ between elements of $S$ and morphisms of $G$, and thus ...
3answers
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### Uniqueness of adjoint functors up to isomorphism

Suppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...
1answer
50 views

### Viewing groups as objects of the concrete category $\mathsf{Grp}$

Sometimes I ask questions about how structures (groups, topological spaces etc.) ought to be defined, and oftentimes a categorial solution is suggested. Here is a recent example. Now from my ...
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82 views

### Basic categories cheat sheet

Has anyone come across a cheat sheet containing basic properties of the most well-known categories (i.e. does it have (co)products, (co)equalizers, (co)limits, etc?)?
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### Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof

What if the set of all well-formed statements in ZFC formed a kind of pseudo-category where a morphism f between objects A, B represented a formal proof that A implied B? What if that category could ...
1answer
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### Awodey - A question about Remark 1.7

At pag 15. Theorem 1.6 Every category C with a set of arrows is isomorphic to one in which the objects are sets and the arrows are functions. The following Remark 1.7 is: "This shows us what is ...
0answers
41 views

### Question about bifunctors/bimodules

I'm wondering how a functor $C \to D$ induces a bifunctor $C \times C^{op} \to D$ (it's an "example"). Am I downright stupid not seeing this? Second problem I have: For categories $C$ and $D$: A ...
1answer
114 views

### A natural example in category theory

I'm looking for a natural example of a category $\mathcal{C}$ with finite limits (or just finite products) wherein some object $X$ is not isomorphic to a subobject of an inhabited object. In other ...
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65 views

### Sheaf as a functor

Let $X$ be any topological space, $S$ - any category (e.g. of sets). Consider a new category $C$: its objects are only open subsets of $X$ and a set of morphisms from $U$ to $V$ is nonempty if and ...
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113 views

### Path Algebra for Categories

For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
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### Existence of not locally small categories

I had a strange remark answered to one of my questions some time ago. My question was involving "locally small categories", and that comment was saying that the existence of not locally small ...
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### Free objects in $\mathrm{Set}(G).$

What are the free objects in the category of $G$-sets for a group $G$? After considerable deliberation (I'm not very bright), I'm pretty sure they are the $G$-sets $X$ on which $G$ acts freely, that ...
3answers
75 views

### Examples of epimorphisms which are not split epimorphisms?

Are there some examples of epimorphisms which are not split epimorphisms? Thank you very much.
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35 views

### How to pose a decent counting problem of cells in higher category?

Suppose you are given a power series $$S = \Sigma_{i=0}^{\infty}a_ix^{i}$$ with coefficients in $\mathbb{N}$, and you are tasked with telling if there can not be a finite category (or any kind of ...
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69 views

### About the category $\mathrm{Set}(G)$

I'm not good with categories. I've attempted several times to understand what a natural transformation is, and so far I've failed. But I'm trying to learn algebraic topology now, and it seems that I ...
1answer
34 views

### Has every set a universal map with respect to the “squaring functor”?

Exercise $26C$ of Herrlich & Strecker Category theory asks to show the following: Show that every set has a universal map with respect to the "squaring functor". Recall that a universal map ...
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125 views

### What about a module of rank $\frac{1}{2}$?

Let $R$ be a commutative ring. The possible ranks of free $R$-modules are $0,1,2,\dotsc$. But what about a generalized notion of an $R$-module where ranks may be rational numbers such as ...
2answers
83 views

### Left Adjoint of a Representable Functor

Let $\mathcal{C}$ be a category with coproducts. Show that if $G:\mathcal{C} \to \mathbf{Set}$ is representable then $G$ has a left adjoint. I can't seem to wrap my head around this nor why ...
1answer
63 views

### How to define topology in terms of subobjects?

How to define topology for general categories. The following is my attempt to do so but I guess it is not correct. What should be the categorical analouge of the third axiom of topology (that the ...
0answers
48 views

### What is the notion for this arrow in category theory?

When reading a CS paper, I encountered an arrow like $\langle f_1, \ldots, f_n \rangle : \langle A_1, \ldots, A_n \rangle \rightarrow \langle B_1, \ldots, B_n \rangle$. The author took it for ...
2answers
63 views

### Thin categories: up to isomorphism Vs up to equivalance

In ncatlab entry for a thin category* reads: "Up to isomorphism, a thin category is the same thing as a proset. Up to equivalence, a thin category is the same thing as a poset" I would very much ...
1answer
78 views

### Left adjoint in a functor category

Edit: Originally put "right adjoint" instead of "left adjoint"; now changed. If I have small categories $\mathcal{C},\mathcal{D}$ with $F : \mathcal{C} \to \mathcal{D}$ a functor then I want to show ...
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40 views

### functor convert disjoint union to direct sum

My question is: How the functor convert disjoint union to direct sum in sense of Mackey functor(Tambara functor). Any help? please. Thank you
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### Existence of a certain functor $F:Grpd\rightarrow Grp$

Let $Grpd$ denote the category of all groupoids. Let $Grp$ denote the category of all groups. Are there functors $F:Grpd\rightarrow Grp, G:Grp\rightarrow Grpd$ such that $GF=1_{Grpd}$. Dear all, I ...
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130 views

### What kind of object is “the product of all objects of a category”?

Let us denote the set of all objects of a small complete category by $C^{\bullet}$. My question is concerned with the limit of the diagram $$C^{\bullet} \longrightarrow C$$ which sends every morphism ...
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### Questions about Rickards proof that $D^b_\mathtt{sg}(A) \equiv \mathtt{stmod}(A)$

Setup: Let $A$ be a self-injective algebra (so projective = injective for modules) and let $D^b(A)$ and $K^b(A)$ be the bounded derived category and the full subcategory consisting of the perfect ...
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### Categorical Confusion in Rotman's Advanced Modern Algebra Second Edition

In Rotman's Advanced Modern Algebra, exercise 6.45 (ii) in the second edition, he gives us objects $X, C_1, C_2$ and morphisms $g_1: X \rightarrow C_1$, $g_2: X \rightarrow C_2$, and asks us to prove ...
1answer
54 views

### distribution of categorical product (conjunction) over coproduct (disjunction)

In the classical and intuitionistic propositional calculi, it is straightforward, using natural deduction, to derive $((A \land C) \lor (B \land C))$ from $(A \lor B) \land C$: Assume \$(A \lor B) ...
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### Morphisms in a category with products

I'm having a hard time proving that $$(\psi\phi)\times(\psi\phi)=(\psi\times\psi)(\phi\times\phi),$$ where $\phi:G\to H$ and $\psi:H\to K$ in some category with products. I have seen a diagram of this ...
0answers
35 views

### Plus construction of sheafification as a colimit of presheaves.

In Sheaves in Geometry and Logic, Moerdijk and Mac Lane construct the associated sheaf on a site $(\mathcal C, J)$ of a presheaf $P$ as $$a(P) = (P^+)^+ ,$$ where $P^+$ is defined pointwise as ...
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### Uniqueness of the operation for a preadditive category?

When working on problems in Rotman's Algebra, he asks us to show that Groups is not a preadditive category. If we could show that the binary operation on $\mathrm{Hom}(A,B)$ had to be \$f + g \mapsto ...
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164 views

### On the large cardinals foundations of categories

(This was cross-posted to MathOverflow.) It is well-known that there are difficulties in developing basic category theory within the confines of $\sf ZFC$. One can overcome these problems when ...
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97 views

### Category theory $\subset$ Set theory or vice versa?

I just started reading the ABC of category theory using the appendix of a text, the first chapter of a text that I have never read, and above all (I found out now that they handle well the theory) the ...
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### Defining Test-Objects

In various categories one encounters what are referred to as 'test-objects'. For example, singleton set serves as test-object (in testing the equality of functions) in the category of sets. In the ...
1answer
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### Does completeness of a category in an adjunction imply completeness for the other?

Assume we have an adjunction $(L,R,\varphi):\mathcal{C}\rightarrow\mathcal{D}$ between two categories, and assume also that $\mathcal{D}$ is complete (i.e. closed under limits). Under what assumptions ...
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