Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions 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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3
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3answers
115 views

Has the opposite category exactly the same morphisms as the original?

This is actually a question about categories; not only about the category that I mention here specifically. I only use category $\mathsf{Rel}$ as an example. How to describe a morphism that ...
0
votes
0answers
24 views

What are the properties of the category of all categories within itself? [on hold]

Let us suppose that we use something like new foundations instead of ZFC for our set theory (any set theory that allows this question to make sense would work.) This allows us to have a category of ...
0
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0answers
13 views

Category theoretic view of coupling measures/RVs

Here is a general definition of the word "coupling" that covers every use I've seen of it. (And this generality is necessary because sometimes one does not define a coupling on an exact product space, ...
2
votes
2answers
50 views

Homotopy direct limit versus direct limit

Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and ...
6
votes
1answer
55 views

The category of theorems and proofs

On a philosophy website, it said that you could have a category with theorems as objects and proofs as arrows. This sounds awesome, but I couldn't find anything on the web that has both "category" and ...
0
votes
1answer
35 views

manipulation of internal hom [on hold]

Let $\text{hom}(-,-):\mathcal{C}^{op}\times\mathcal{C}\rightarrow\mathcal{C}$ denote the internal Hom functor associated to a closed category $\mathcal{C}$, and let $X$, $Y$, and $Z$ be objects in ...
1
vote
1answer
43 views

Standard Notation For The Set of All the Morphisms Of A Category

Let $\mathscr C$ be a category. Let $\text{Ob}(\mathscr C)$ be the set of all the objects of $\mathscr C$. Is there a standard notation for $\bigcup_{A,B\in\text{Ob}(\mathscr C)}\text{Mor}(A,B)$? ...
11
votes
1answer
198 views

Joke explanation: “a comathematician is a device for turning cotheorems into ffee”

Ok, so apparently there is an old joke (which I DO get) that says that in Hungary a mathematician is a device for turning coffee into theorems. I found a post by Qiaochu Yuan that has the following ...
3
votes
0answers
29 views

What are the algebraic and topological properties of a tree (graph theory), and what are any possible connections?

I'm a non-mathematician working with trees (mostly rooted and oriented trees). Typically, I understand them as join-semilattices, so they have an implicit algebraic structure: they form a subgroup ...
3
votes
1answer
49 views

What's the difference between a logic, an internal logic (language) of a category, an internal logic of a topos and a type theory?

maybe this question doesn't make sense at all. I don't know exactly the meaning of all these concepts, except the internal language of a topos (and searching on the literature is not helping at all). ...
2
votes
0answers
17 views

Does the lax Gray tensor product preserve fully faithful 2-functors?

The question is in the title. By fully faithful 2-functor, I mean 2-functors such that the maps on the hom categories are isomorphism, and by preserve, I mean in each variable. I have an argument ...
1
vote
0answers
43 views

Epimorphisms in two directions

Let $\mathcal{C}$ be a category. Consider the following statement: (S1) Whenever $A,B$ are objects and $\iota_1:A\to B$ and $\iota_2:B\to A$ are monomorphisms, then there is an isomorphism $\phi:A\to ...
5
votes
0answers
65 views

Is Category Theory geometric?

In "From a Geometrical Point of View" (http://www.amazon.com/gp/aw/d/1402093837?pc_redir=1407132421&robot_redir=1) Marquis states that category theory is thoroughly geometric. Could someone ...
1
vote
1answer
40 views

Additive, covariant functor commutes direct limits, then it commutes with direct sums?

Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ ...
4
votes
0answers
83 views

Learning roadmap to Topological Quantum Field Theories from a mathematics perspective

I want to learn TQFT's and am looking for review articles or books. My mathematics knowledge is limited to one year of graduate course in Algebra (Groups,Rings,Fields,Categories, Modules and ...
1
vote
1answer
26 views

Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
3
votes
2answers
49 views

Help with exercise from the Category Theory Wikibook

Reading through the Category Theory Wikibook, I came across the following exercise: (Harder.) If we add another morphism to the above example, it fails to be a category. Why? Hint: think about ...
0
votes
0answers
48 views

Relation between existential and universal quantificator in category theory

Let $\mathscr C$ be a cartesian (i.e. with finite limits) category with subobject classifier $\Omega$ and generic subobject $\tau:I\to\Omega$ (here $I$ denote the terminal object). Let $f:X\to Y$ and ...
2
votes
0answers
37 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
1
vote
1answer
39 views

Every closed (not-necessarily symmetric) monoidal category is canonically self-enriched, right?

Here it is stated that: A closed symmetric monoidal category is canonically self-enriched. This makes sense, but I don't see why it has to be symmetric. Every closed (not-necessarily symmetric) ...
1
vote
0answers
40 views

Is this a fruitful enrichment of $R[X]$?

Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a ...
1
vote
1answer
53 views

Exponential objects and hom-sets.

Let $C$ be a cartesian closed category and $X, Y$ two objects of $C$. Is it the case that $\text{Hom}(X,Y) = Y^X$?
3
votes
0answers
75 views

Showing that morphisms transforms as claimed, Borceux and enriched natural transformations.

I am having trouble following the proof given in the images below of lemma 6.3.3. More specifically, it is claimed that diagram 6.22 is equivalent to diagram 6.23, but I can't see it. Here, the object ...
2
votes
1answer
36 views

Derived Functors and nice Resolutions

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following: "Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor ...
4
votes
1answer
40 views

“Nice proof” that the unit of the left Kan extension of $F$ is an isomorphism, if $F$ is fully faithful

Let $F: \mathbf C \to \mathbf D, G: \mathbf C \to \mathbf E$ be functors. Assume that $\mathbf C$ is small, $\mathbf D$ is locally small and $\mathbf E$ is cocomplete. Then, I can compute the left Kan ...
0
votes
2answers
41 views

A property of product category

This property of the product category states that the projections $P$ and $Q$ are "universal" among pairs of functors to $B$ and $C$. Can someone specify me exactly the sense of that assertion? I ...
0
votes
0answers
33 views

directed colimits,category,unary algebras,preservation

Are there some natural properties of mono-unary algebras NOT preserved by (omega)-directed colimits (unlike being connected,having no cycles...)? Formalization to logical formulas is not necessary.
1
vote
0answers
22 views

Map of monads and left adjoints

Let $(T,\eta,\mu)$, $(T',\eta',\mu')$ be two monads on a category $X$. Let $\theta:T\Rightarrow T'$ be a map of monads. Then, we have an induced functor $X^\theta:X^{T'}\rightarrow X^T$ (from the ...
5
votes
1answer
80 views

Category theory for knot theory

Where would I start to learn category theory for its use in knot theory? I have a background in physics and Ive read Adams Knot book. I know nothing about category theory. Eventually I want to learn ...
2
votes
1answer
45 views

Terminal objects of the category of morphisms

I'm reading Basic Category Theory for Computer Scientists by Benjamin C. Pierce and in exercice 1.4.6, he asks what the terminal objects are in $Set^\to$. Let $C$ be a category. The category ...
4
votes
1answer
84 views

Composition of bicartesian squares

A commutative square is called bicartesian when it is both pull-back and push-out. In an abelian category, consider two pull-back squares $(X)$ and $(Y)$: $$ \begin{array}{ccccc} A & ...
0
votes
1answer
29 views

Pos,coequalizer,terminal poset

Consider the reflexive pair $u, v \colon (1 + 2) \to 2$ from the coproduct of the terminal poset $1$ and the two-element chain, where both morphisms are the identity on the second summand, and they ...
1
vote
1answer
47 views

Free object in category of groups.

Suppose $X$ is a set and $F$ is a free object on $X$ (with $i:X\rightarrow F$) in the category of groups. Prove that $i(X)$ is a set of generator for the group $F$. I have the following hint: If ...
1
vote
1answer
102 views

Are tensor products of vector bundles “well-behaved”?

Do the "nice" properties of the tensor product of vector spaces always extend to tensor products of vector bundles? I'm working through Milnor-Stasheff and recently had to prove that the tensor ...
5
votes
0answers
74 views

ind-completion and functors which are full with respect to isomorphisms

Let $C$ and $D$ be categories and $F:C\rightarrow D$ a faithful functor which is full with respect to isomorphisms. This means that if $a,b\in C$ and $f:F(a)\rightarrow F(b)$ is an isomorphism in $D$, ...
2
votes
0answers
44 views

Right Kan extension along a diagonal functor.

Denote $\newcommand{\simpcat}{\boldsymbol\Delta} \simpcat$ the simplex category (category of finite ordinals with non decreasing maps). The diagonal functor $\delta \colon \simpcat \to ...
3
votes
1answer
37 views

Adjunction counit for sheaves is isomorphism

Let $f\colon X \to S$ be a proper morphism of varieties over $\mathbb{C}$ with $f_* \mathcal{O}_X = \mathcal{O}_S$ and $\mathcal{G}$ be a coherent sheaf on $S$. Then we have a natural morphism ...
2
votes
1answer
63 views

The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
1
vote
0answers
41 views

Size of Hom-Sets in A Functor Category

I am trying to prove the following presumably easy fact: if $B$ is a category with small hom-sets and $C$ is a small category, then $B^{C}$ has small hom-sets. I am assuming the standard Z-F axioms ...
3
votes
2answers
103 views

Natural Transformations Without Objects

So I've been thinking about the definition of categories as just arrows with a defined composition (i.e. without objects). I understand this is silly, but it's fun and I have a question about it: ...
2
votes
1answer
102 views

Every Abelian group is canonically a $\mathbb{Z}$-module. Is this just a coincidence?

Every Abelian group is canonically a $\mathbb{Z}$-module, where $\mathbb{Z}$ is the initial monoid in the monoidal category of Abelian groups. And every Abelian monoid is canonically an ...
1
vote
2answers
204 views

why are the category of pointed sets and the category of sets and partial functions “essentially the same”?

I'm reading "an introduction to category theory" by Harold Simmons. In this books, exercise 1.2.7 wants us to show that $\mathcal{Set}_\bot$ (the category of pointed sets) and $\mathcal{Pfn}$ (the ...
0
votes
1answer
43 views

how would you define the term “elementary” in the context of categories and sets?

I was just reading P.T. Johnstone's introduction to his book "Topos Theory", where he uses the term "elementary" many times to classify the nature of theorems and definitions, examples below. I ...
0
votes
0answers
23 views

example of algebraic theory,free product completion,graphs

Let us denote by $\def\Graph{{\sf Graph}}\Graph$ the category of directed graphs $G$ with multiple edges: they are given by a set $G_v$ of vertices, a set $G_e$ of edges, and two functions from $G_e$ ...
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vote
0answers
59 views

Generalizing a statement about direct limits in the category of $A$-modules to other categories

The original problem is from Atiyah and Macdonald's Introduction to Commutative Algebra, Ex 2.15: Suppose $(M_\alpha,\mu_{\beta\gamma})$ is a direct system of modules over a unital commutative ...
0
votes
0answers
73 views

Tensor product of arbitrary categories

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...
0
votes
1answer
113 views

How to prove that : $ \mathrm{Hom} ( A(G), H) \simeq \mathrm{Hom} (G , I(H)) $?

How do we show that the functor $ A : \mathrm {Gr} \to \mathrm {Ab} $ defined by $ A (G) = G / [G, G] $ is a left adjoint functor of the inclusion functor : $ I : \mathrm {Ab} \to \mathrm {Gr} $ ?. ...
0
votes
0answers
41 views

surjection between sets which are defined through a functor

I'm facing the following problem and have no idea how to deal with it. We consider a functor $T:\underline{Set}\rightarrow\underline{Set}$ and two sets $X,Y$. We can build the product $X\times Y$ ...
1
vote
0answers
42 views

Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
1
vote
2answers
84 views

What would be the equivalent of the “gluing axiom” for a cosheaf

A sheaf is a presheaf $F$ such that for all $U$ and for all covering $\{U_i\}_{i\in I} $ of $U$, $F(U)$ is the equalizer $$ F(U) \overset{f}{\longrightarrow}\prod_{i\in I} F(U_i) ...