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 ...
7
votes
1answer
62 views
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 ...
6
votes
1answer
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 ...
12
votes
4answers
132 views
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. ...
3
votes
1answer
74 views
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 $H$. ...
1
vote
1answer
25 views
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 ...
1
vote
1answer
29 views
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 ...
10
votes
1answer
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 ...
4
votes
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 ...
3
votes
1answer
35 views
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 = ...
1
vote
1answer
39 views
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.
...
2
votes
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 ...
1
vote
1answer
31 views
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 $\mathcal{D}$ ...
5
votes
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 ...
0
votes
1answer
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 ...
1
vote
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 ...
4
votes
1answer
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 ...
3
votes
3answers
66 views
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 ...
1
vote
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 ...
9
votes
0answers
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?)?
4
votes
0answers
57 views
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 ...
1
vote
1answer
50 views
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 ...
1
vote
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 ...
8
votes
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 ...
4
votes
1answer
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 ...
10
votes
2answers
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", ...
2
votes
2answers
83 views
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 ...
4
votes
2answers
69 views
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 ...
2
votes
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.
2
votes
0answers
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 ...
4
votes
2answers
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 ...
0
votes
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 ...
8
votes
2answers
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 ...
5
votes
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 ...
4
votes
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 ...
0
votes
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 ...
1
vote
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 ...
3
votes
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 ...
-1
votes
0answers
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
9
votes
0answers
90 views
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 ...
4
votes
3answers
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 ...
10
votes
2answers
92 views
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 ...
2
votes
1answer
56 views
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 ...
4
votes
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) ...
4
votes
2answers
77 views
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 ...
1
vote
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 ...
4
votes
4answers
76 views
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 ...
10
votes
0answers
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 ...
3
votes
3answers
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 ...
1
vote
1answer
46 views
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 ...
5
votes
1answer
33 views
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 ...
