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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227 views

Will learning category theory lead to a better and clearer understanding of mathematics?

I read the first chapter on a book about category theory Conceptual Mathematics:A first introduction to categories.In the preface the authors say: It has been the good fortune of the authors to live ...
4
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1answer
84 views

Do any familiar adjunctions arise from this construction?

I was pondering an answer that user Andreas Blass provided to an old question of mine and wondered if the following construction cropped up anywhere other than the example I will provide. For any set ...
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44 views

Check that this is a category

Assume we have a fixed field $F$. We define objects as homomorphisms $\phi:F\rightarrow G$. Then we define morphisms between $\phi:F\rightarrow G$ and $\psi:F\rightarrow L$ as ring homomorphism from ...
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71 views

Categories in which coproducts embed into products

Let $\mathcal{C}$ be a category with coproducts and zero morphisms. Then we have projections $\bigoplus_{i \in I} M_i \to M_i$. For every object $T$ they induce a map $\hom(T,\bigoplus_{i \in I} M_i) ...
5
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1answer
216 views

Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following ...
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2answers
119 views

Category In Which Not All Free Objects Exists

I am trying to think of a category in which not all free objects exists. I thought this might be the case in sets (I thought I might be able to violate the uniqueness ) but I couldn't get anywhere so ...
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1answer
45 views

Example of a functor on products

I am trying to come up with an example of a functor which maps products to products but not the same one. That is: Let $C,D$ be categories such that $C$ has all products. I need to define ...
4
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1answer
46 views

Why is sets not a path category

If we let $U$ be a class and let disjont sets $Q(A,B)$ be given for each pair $(A,B)\in U^2$ then the associated path category is the category $\mathfrak{C}$ with: $ob(\mathfrak{C})=U$ and morphisms ...
3
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1answer
152 views

Is the continuation monad terminal?

For each $R$ an object of a cartesian closed category, there is a monad $\mathrm{Cont}_R(A) = [[A,R],R]$, the continuation monad. If $M$ is a strong monad, we can find for each pair of objects $A$ ...
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2answers
91 views

Locally small category whose collection of isomorphism classes cannot be a set

For natural examples of locally small categories like the category $\mathbf{Grp}$ of groups, the isomorphism classes themselves are normally not sets. In a set theory like ZFC, even the collection of ...
3
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1answer
113 views

Is every “almost” isomorphism an isomorphism?

Let $f:A \mapsto B$, $g:B \mapsto A$ and $h:B \mapsto B$ be such that $g \circ f=\operatorname{id}_A$ and $f \circ g \circ h=\operatorname{id}_B=h \circ f \circ g$. Can we conclude ...
2
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1answer
77 views

Is possible to define the parity by a universal property?

Consider the parity homomorphism of the symmetric group $$ p:S_n\to Z/(2). $$ Is it possible to characterise this map by a pure universal property? This question occurred to me when I was reading ...
2
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2answers
204 views

Why is currying part of the definition of exponential objects?

For context: an exponential of objects $B$ and $A$ in a category $\mathcal{C}$ is defined as an object $B^A$ and a morphism $\epsilon: B^A \times A \rightarrow B$ such that for every object $C$ in ...
3
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0answers
62 views

An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
2
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0answers
41 views

Pullback in specific category

can someone help me with the following Let $C$ be the category with objects subsets of $\mathbb{N}$, and arrows functions $f:A \to B$ such that preimage of each point is a finite set i.e. for every ...
2
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1answer
59 views

A very simple question on adjunctions that I simply can't look at correctly.

Suppose that $\mathcal{C}$ has finite products. Why is the unit $\eta_c:c\to c\times c$ of the adjunction $\Delta\dashv \times:\mathcal{C}\times\mathcal{C}\to\mathcal{C}$ the arrow $\langle ...
4
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1answer
69 views

Proving things about the free group from the categorical definition.

I'm undertaking A Course in the Theory of Groups by Robinson and I'm looking for some guidance on some of the exercises. Specifically, I'm trying to show that a free group of rank 2 or higher has a ...
7
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1answer
190 views

Preserving tensors and cotensor and adjoint functors

Let $\mathcal{V}$ be a symmetric monoidal category and $\underline{\mathcal{M}}$ and $\underline{\mathcal{N}}$ be cotensored and tensored $\mathcal{V}$-categories. Now, say that we have an adjunction ...
4
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1answer
148 views

Cokernels in the category of free abelian groups

My question is if there are Cokernels in the category of free abelian groups. The answer is yes in the case of finitely generated free abelian groups since one has the structure theorem of finitely ...
9
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4answers
247 views

Is there a “natural” / “categorical” definition of the “parity” of a permutation?

Given a permutation $\sigma$ on $n$ elements (i.e. $\sigma \in S_n$), there is a notion of "parity" (or "sign" or "signature") of $\sigma$, which can be defined in several equivalent ways (look here). ...
4
votes
1answer
236 views

Does localization functor have both sides adjoint functors?

Let $A$ be commutative ring, and $S$ a multiplicative set. The localization $S^{-1}$: $A$-module $\rightarrow$ $S^{-1}A$-module. Functor $F$: $S^{-1}A$-module $\rightarrow$ $A$-module, regard $A$ as ...
0
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1answer
58 views

What's the name of this category

We can usually build new categories from old ones, as example we have the slice $\mathfrak C/A$ and coslice categories $A/\mathfrak C$ of $\mathfrak C$ with an object $A$. I'm reading this book and ...
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0answers
52 views

Relationship between groupoid morphisms and the induced functor on their categories of actions?

This is similar to a question I asked recently, but this time specifically for groupoids. Suppose $f: A \rightarrow B$ is a groupoid morphism. Let $f^\ast: [B, \text{Set}] \rightarrow [A, ...
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0answers
34 views

Given an arrow from a category to a product of categories, can we evaluate it on an object using its unique decomposition?

Suppose you have a morphism $f:\mathcal{C}\to \mathcal{D}\times\mathcal{E}$ of categories. Then $f=\langle \pi_{\mathcal{D}}\circ f,\pi_{\mathcal{E}}\circ f\rangle$, uniquely, by the definition of the ...
2
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1answer
188 views

Conditions to ensure the chain homotopy category $K(\mathcal{A})$ is abelian?

It is known that the chain homotopy category $K(\mathcal{A})$ for an abelian category $\mathcal{A}$ need not be abelian. For example, $K(\mathrm{Ab})$ is not even abelian. Are there any known ...
3
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1answer
121 views

Are all left exact functor are right adjoint functors?

Right adjoint functor is left exact. I am curious about the inverse statement. Thanks!
3
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1answer
83 views

categorification and linear algebra

Vect is the category with objects as vectors and arrows as linear transformations between them. Then these arrows have quite a bit of structure. We can take the transpose, trace, determinant, ...
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1answer
140 views

What is the relationship between the second isomorphism theorem and the third one in group theory?

The second isomorphism theorem [wiki] in group theory is as follows: Let $G$ be a group. $H \triangleleft G, K \le G$. Then: $HK \le G$, $(H \cap K) \triangleleft K$, and $K/(H \cap K) ...
2
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3answers
104 views

Is this a correct characterisation of natural equivalence?

In the category of functors of type $\mathcal{C}\rightarrow \mathcal{D}$ an arrow $\psi: F \Rightarrow G$ is defined as a collections of $\left(\psi_X: FX \rightarrow GX| X:\mathcal{C} \right)$ of ...
4
votes
1answer
75 views

cokernel pairs left adjoint to equalizers

Given a category $ C $ which has both cokernel pairs and equalizers, how can I see that the functor $ C^\downarrow\longrightarrow C^{\downarrow\downarrow} $, which takes every arrow in $ C $ to its ...
3
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1answer
54 views

Proving naturality of an isomorphism in MacLane's CWM.

I am going through the adjuntions chapter of MacLane's CWM. I will follow his notation; it would be very difficult to try to describe all of the notation here. So excuse me for not explaining ...
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1answer
46 views

Basic localizers contain adjoint functors

I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences. Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category ...
2
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1answer
50 views

Cylinder with bases collapsed to a point.

The problem, although arising from some deeper facts, is quite simple. I would like to visualise the quotient space $A$ given by the cylinder $I\times S^{1}$ ($S^{1}$ is the circle in $\mathbb{R}^{2}$ ...
1
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1answer
59 views

Arrow between endofunctors over a symmetric monoidal category.

Consider an arrow between the categories of endofunctors over two symmetric monoidal (SM) categories $\mathcal{C}$ and $\mathcal{D}$ $$a:End(\mathcal{C}) \rightarrow End(\mathcal{D})$$ It is a ...
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0answers
52 views

exercise involving exactness

I have been stuck on this exercise for a little while. We are in abelian category. How do I show that $0 \rightarrow A \rightarrow B$ is exact if and only if $f: A \rightarrow B$ is a monomorphism? ...
3
votes
2answers
99 views

Which constructions on a category are still interesting for a groupoid?

By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ...
3
votes
1answer
204 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
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0answers
118 views

Category of Presheaves on a small category $C$ is locally cartesian closed

I'm trying to fill in the details of the proof and need the following result: $Set^{C^{op}}/P\simeq Set^{D^{op}}$, where $D$ is the category of elements of $P$. The objects are pairs $(x,C)$ with $C ∈ ...
2
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1answer
72 views

equivalence in category

First gives some definitions, and then the property that I am confused. $A$, $B$ are both $R$-module, and $C$, $D$ an (additive) abelian group, consider the category $M(A,B)$ whose objects are all ...
2
votes
1answer
144 views

Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
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2answers
71 views

What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
2
votes
1answer
37 views

Behaviour of $\operatorname{Ext}$ with left exact sequences.

Maybe is a trivial question but I am not so good in derived functors. Assume we are in the category of abelian groups and we have an exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow ...
1
vote
1answer
64 views

Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
2
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2answers
317 views

Foundations book using category theory for student embarking on PhD in mathematical biology?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
3
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1answer
92 views

Topological construct

I just started working with some category theory and I would like to understand the link between what I am studying now and what I know about topological spaces. By definition, a construct (in our ...
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0answers
68 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
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1answer
92 views

Underlying functor of tensor product in a closed and symmetric monoidal category.

I will follow, for terminology and notation, G. M. Kelly, Basic Concepts of Enriched Category Theory. For sake of a self-contained exposition, I will try to write here all the needed concepts. Let ...
5
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2answers
196 views

Is there a category of categories?

My question is quite simple, I would like to know if we can define the category of the categories, unlike Cat which is the category of the small categories. By the way, are there any particular reason ...
3
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1answer
135 views

Properties Shared by Equivalent Categories

If two categories are equivalent, then if one has products, then so does the other. The proof of this is easy enough so I'm guessing the same result holds for exponentials but I am having trouble ...
3
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1answer
74 views

The idea of “generators” for arbitrary categories

Given a partial order $\langle X, \leq \rangle$ and a subset $I \subseteq X$ it is common to consider $I$ as the generators of the set $\{ x \in X: i \leq x \textrm{ for some }i \in I \}$ (i.e., the ...