Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seeming different areas of mathematics.

Filter by
Sorted by
Tagged with
0
votes
2answers
36 views

If $\mathscr A$ has all products and equalizers, then it has all limits

This question is about part (a) of this proposition: Here is a plan of the proof. Here's what the picture looks like, from what I understand: But I don't understand what the condition $s\circ p=t\...
3
votes
1answer
47 views

The set $5^{-\infty}\mathbb{Z}$ is a colimit

I am trying to understand why the set of rational numbers whose denominators are powers of $5$, $5^{-\infty}\mathbb{Z}$, is a colimit. Specifically, why is $5^{-\infty}\mathbb{Z}$ the colimit of the ...
0
votes
0answers
26 views

The gist of Section II.3: “Sheaves and Manifolds” in Mac Lane and Moerdijk.

I'm reading Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]" for fun. It says readers may skip Section II.3 on Sheaves and Manifolds. Since I have very little experience with ...
0
votes
1answer
40 views

How to construct a short exact sequence of complexes

Suppose that a hsort exact sequence $$ 0 \longrightarrow A \overset{f}{\longrightarrow B} \overset{g}{\longrightarrow} C \longrightarrow 0 $$ of objects in some (Abelian) category is given. Also, ...
3
votes
0answers
24 views

A ring for which the contravariant hom functor (dual functor) is conservative.

Is there any condition that guarantees that a commutative ring $R$ satisfies the condition that a contravariant hom functor ${\rm Hom}_R(-,R):{\rm Mod}_R^{op} \to {\rm Mod}_R$ is a conservative ...
0
votes
2answers
52 views

Injective vs. monic (in categories where it makes sense)

The question is about the example (from here): 1) It appears to me from the proof of the fact that injections are monic that this fact is true for any category whose objects are sets (possibly with ...
7
votes
0answers
59 views

Algebraic Geometric Analogue of Brown's Representability

Brown's representability theorem is very usefull to show that the functor $$X \rightarrow H^i(X,A)$$ is representable. I would be interested to see if there exists an analogue of this statement in ...
2
votes
1answer
58 views

Part of proof of Yoneda's Lemma from Vakil

I am trying to understand the second half of the proof of Yoneda's Lemma, which is given as a problem in Vakil's notes. So suppose we have two objects $A$ and $A'$ in a Category $D$, and morphisms $...
-1
votes
0answers
12 views

Any two chain-homotopic maps of chain complexes induce the same maps on homology [closed]

How do I prove that any two chain-homotopic maps of chain complexes induce the same maps on homology?
3
votes
0answers
45 views

Filling a map $X\times\Lambda^n_k\rightarrow Y$ to a map $X\times\Delta^n\rightarrow Y$

I've come across the following claim here, in lemma (1.15): if $X$ and $Y$ are simplicial sets and $Y$ is a Kan complex, then for any $k\leq n$ and $\varphi:X\times\Lambda^n_k\rightarrow Y$ there is a ...
1
vote
0answers
44 views

A variation of a codomain fibration

Let $\mathcal C$ be a category with pullbacks, let $Arr(\mathcal C)$ be its arrow category. The functor $cod\colon Arr(\mathcal C)\to\mathcal C$ that takes an object of $Arr(\mathcal C)$, i.e. a ...
4
votes
2answers
102 views

What lies behind the definitions of split monics and epics?

Is there an easy way to memorize the definitions of split monics and split epics, and not to confuse the domains/codomains of the arrows from those definitions? For example, is there a mnemonic rule?...
2
votes
1answer
67 views

What is the corresponding categorical notion to a non-functorial modality?

Consider the constructive logic with a modality with the following modal axioms: □(a → b) → □ a → □ b ...
2
votes
0answers
39 views

Preservation of weak pullback

A weak pullback is defined in the same way as a pullback, but the arrow to the vertex of the limit cone is not required to be unique. Here's the problem: Let $\mathscr P:\mathbf {Set}\to\mathbf{...
2
votes
0answers
37 views

An arrow is monic iff the square is a pullback

Here's a lemma: Is the following proof correct? Suppose the square is a pullback. Then for all objects $Z$ and arrows $\alpha,\beta: Z\to X$ such that $f\alpha=f\beta$, there is a unique $\Gamma:Z\...
1
vote
1answer
34 views

Adjoints are Kan extensions

How do I prove that if $F:\mathcal{C}\leftrightarrows\mathcal{D}:G$ is an adjunction with unit $\eta:id_{\mathcal{C}}\Rightarrow GF$ and counit $\epsilon: FG\Rightarrow id_\mathcal{D}$, then $(G,\eta)$...
2
votes
1answer
58 views

Proving Proposition I.5.1 of Mac Lane and Moerdijk.

This is Exercise I.11 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: Adapted from p. 25, ibid. . . . Definition: Let $\mathbf{C}$ be a category. Then $\hat{\...
8
votes
0answers
86 views

What objects can be turned into a category?

A poset $(P,\le)$ can be turned into a category in a standard way. A group $(G,\cdot)$ can also be turned into a category in a standard way. Can we topological space $(X,\tau)$ into a category in a ...
1
vote
1answer
45 views

Idempotent completion

It is claimed in idempotent completion nlab page that the any (co)complete category is idempotent complete. How is this true? My guess given an idempotent $(A,e)$, $e^2=e$, we cavn define the ...
0
votes
1answer
19 views

$Et/X$ is closed under finite limits in $Top/X$.

Let $X$ be a topological space and $Et/X$ the full subcategory of $Top/X$ of local homeomorphisms. The question is as in the title. Let $(L \xrightarrow{l} X, \mu)$ be a limiting cone for $M: I \to ...
0
votes
0answers
29 views

Generalise Theorem 2 of Section 9 of Part I of “Sheaves in Geometry and Logic [. . .]” to presheaf categories.

This is Exercise I.10 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: From p. 17 ibid. . . . Definition: Given two functors $$F:\mathbf{X}\to \mathbf{A}\...
1
vote
1answer
46 views

In $\mathbf{Sets}^\mathbf{Q}$, prove the subobject classifier $\Omega$ is given by $\Omega(q)=\{r\mid r\in\mathbf{R^+},r\ge q\}.$

This is Exercise I.9 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". The Details: A definition of a subobject classifier is given on page 32, ibid. Definition: In a category $...
1
vote
1answer
29 views

A tower of Cartesian Products is Cartesian

I am trying to figure out the following exercise from Vakil's notes: If the two squares in the following commutative diagram are Cartesian diagrams, then the "outside rectangle" (involving U,V,Y, and ...
1
vote
2answers
71 views

A project work on algebraic topology (with categorical flavour) : suggestions for topics.

As a part of my exam on Algebraic Topology, I have to prepare a brief exposition deepening a topic treated in the course. The background is: basic homotopy theory (fundamental group, theory of ...
0
votes
0answers
22 views

Names of different products of functions

Are there standard (or at least any non-standard) names for these "products" of functions? $(f\times g)(x) = (f(x),g(x))$; $(f\times g)(x,y) = (f(x),g(y))$.
2
votes
1answer
51 views

Proving that this set is a limit in $\mathbf{Set}$

As a continuation of this question, I'm trying to prove that $$L=\{(x_I)_{I\in \mathbf I} : x_I\in D(I)\text{ for all } I\in\mathbf I \text { and } (Du)(x_I)=x_J \text{ for all } u:I\to J \text{ in } \...
2
votes
1answer
54 views

Proving that $p_I\circ h=p_I\circ h'$ implies $h=h'$

An exercise from Leinster: I was wondering if my solution is correct? (a) Assume $h,h':A\to L$ are arrows such that $p_I\circ h=p_I\circ h'$ for all $I$. Define $f_I=p_I\circ h$, as shown on the ...
0
votes
0answers
60 views

Show that $T^S$ with an evaluation is indeed the exponential in the functor category $\hat{\mathbf{C}}=\mathbf{Sets}^{\mathbf{C}^{{\rm op}}}.$

I'd like to apologise in advance for the mixture of different notations. This is Exercise I.8 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, this is new ...
0
votes
1answer
39 views

The forgetful functor $U:\mathbf{B}G\to\mathbf{Sets}$ need not preserve infinite limits.

This is Exercise I.7 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". Here $\mathbf{B}G$ is the category of all continuous $G$-sets, where $G$ is a topological group. The ...
2
votes
1answer
36 views

Is the cone of the zero map $A \to B$ always $A[1] \oplus B$?

Let $\mathcal{D}$ be a triangulated category, with objects $A, B \in \mathcal{D}$. Is it true that $$A \xrightarrow{0} B \to A[1] \oplus B \to A[1] \tag{$*$}$$ is a distinguished triangle? If $\...
0
votes
0answers
20 views

Example of splitting a fibred category.

Given a fibred category $\mathcal{F} \to \mathcal{C}$, we can choose a cleavage, which is a class of cartesian arrows $K$ in $\mathcal{F}$ s.t. for each arrow $f:U\to V$ in $\mathcal{C}$ and each ...
0
votes
1answer
30 views

Axioms of a Local Set Theory

So this one’s from J.L. Bell’s Toposes and Local Set Theories. Taking a look at the axioms, I’m struggling to see why any sequent of the form$\emptyset$:t would be of much use... Take the axiom of “...
1
vote
1answer
37 views

Subobject of a product

In $\mathbf{Set}$, if $A$ and $B$ are nonempty sets, and $B'$ is a proper subset of $B$, then $A\times B'$ is a proper subset of $A\times B$. Is this true in any topos which is not degenerate? I mean, ...
0
votes
1answer
24 views

General description of colimits in $\mathbf{Set}$ - 2

I've previously posted a question about the example below, but this question is different. Example 5.2.16. The colimit of a diagram $D \colon \mathbf{I} \to \mathbf{Set}$ is given by $$ \lim_{...
0
votes
1answer
46 views

If a functor creates limits, then it also reflects them

Exercise 5.3.10 from Leinster: Prove that if a functor creates limits, then it also reflects them. Here's what I have so far (the question is at the end): Let $F:\mathscr A\to \mathscr B$ be a ...
0
votes
0answers
16 views

Forgetful functors from categories of algebra create limits?

After Leinster states the following lemma (on p. 139): Let $F:\mathscr A\to\mathscr B$ be a functor and $I$ a small category. Suppose that $\mathscr B$ has, and $F$ creates, limits of shape $I$. ...
0
votes
0answers
33 views

If $\mathscr B$ has, and $F$ creates, limits then $\mathscr A$ has, and $F$ preserves, limits

Exercise 5.3.12 in Leinster's book is to prove this statement: Let $F:\mathscr A\to\mathscr B$ be a functor and $I$ a small category. Suppose that $\mathscr B$ has, and $F$ creates, limits of shape ...
2
votes
0answers
16 views

Expectations in the category of finite stochastic maps

I'm learning category theory and trying to boost my intuition by applying what I learn to the category of finite stochastic maps (FinStochMap, defined below). FinStochMap seems a natural category for ...
2
votes
0answers
50 views

how to understand what the inverse limit is? [duplicate]

I really can't understand the meaning of inverse limit and even examples of computing it in diffrent cases and I'm looking for some easy(if such a thing exists!!!) for it. thanks for any help...
1
vote
0answers
48 views

How does the functor Sub act on representable Presheaves?

Let $\mathbb{C}$ be a small category and Psh($\mathbb{C}$) the category of presheaves over $\mathbb{C}$. I just want to consider representable presheaves. The funtor Sub maps a representable presheaf ...
1
vote
1answer
26 views

Are “free products” just coproducts in categories admitting presentations of objects (generators and relators)?

For groups, the "free product" can be taken "generator-wise" and "relator-wise" as done here: https://ncatlab.org/nlab/show/free+product+of+groups It is also the case that the "free product" is the ...
0
votes
1answer
37 views

Conclude that $\mathbf{B}G$ is a CCC using previous questions in “Sheaves in Geometry and Logic [. . .]”.

This is Exercise I.6(c) of the titular book. According the Approach0, this question is new to MSE. The Setup: Let $G$ be a topological group and $\mathbf{B}G$ the category of continuous $G$-sets. ...
1
vote
1answer
64 views

Polynomial Rings “Most Efficient” Ring?

I'm currently reading though Aluffi's Algebra 0, and within III.2.2 Aluffi shows how $\mathbb{Z}[x_1,\dots,x_n]$ satisfies a universal property like which free groups satisfy for groups. Specifically,...
0
votes
0answers
45 views

Simple injective groups [closed]

I would like to know why here in the example $2.5$ the only $\cal H$-injective group is $\{1\}$.
5
votes
1answer
52 views

Uniform probability distributions in the category of stochastic maps

This question is mostly for the purpose of learning how to reason in category theory. Consider the category of finite stochastic maps. I imagine this is something fairly standard, but I've given a ...
1
vote
1answer
69 views

Curry-Howard: Types with Logic vs Types as Logic

In the paper Knowledge Representation in Bicategories of Relations there is a short remark on p47 that makes a distinction that seems quite far ranging regarding the Curry-Howard Correspondence. The ...
1
vote
1answer
48 views

Equivalent definitions of preserving limits

On page 137, Leinster gives two equivalent characterizations of limit preservation: Is it supposed to be obvious that they are the equivalent? If so, how to see that? (When I tried to prove that, I ...
0
votes
1answer
61 views

General description of colimits in $\mathbf{Set}$

I'm not sure I can match the statement given here (from https://arxiv.org/abs/1612.09375) with the real results: Example 5.2.16. The colimit of a diagram $D \colon \mathbf{I} \to \mathbf{Set}$ is ...
3
votes
5answers
976 views

How do we know if two objects in a category are the same or not?

I am wondering whether the category of sets, $\mathbf{Set}$, and the category of groups, $\mathbf{Grp}$, are well defined or not. Suppose we choose a singleton $\{a\}$ from $\mathbf{Set}$. And ...
4
votes
2answers
77 views

Partial order on Grothendieck group of an abelian category

In this article the authors define in 4.1 the Grothendieck group $\mathscr{G}(\mathcal{C})$ of an skeletally small abelian category $\mathcal{C}$ (skeletally small means that the class of isomorphism ...

1 2 3 4 5 198