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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1answer
64 views

Group Duality with respect to Generators and Relations

Although the following question is not phrased in the most accurate way, I would like to ask it in the same way it rushed to my mind: "Looking at some basic examples of group theory with geometrical ...
2
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0answers
51 views

What kind of objects are both subobjects and quotients?

Fix an object $B$ in some category. What does the existence of a diagram $A \rightarrowtail B \twoheadrightarrow A$ imply about $A$ and $B$? What if $A \rightarrowtail B \twoheadrightarrow A$...
5
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1answer
62 views

What can we say if $A\twoheadrightarrow B$ and $A \rightarrowtail B$?

In some category, suppose there are two objects $A$ and $B$ such that the arrow-class $\mathsf{Hom}(A,B)$ has a monic $A \rightarrowtail B$ and an epic $A\twoheadrightarrow B$. Can we say anything ...
1
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0answers
50 views

A hierarchy of arrows by “monicity/epicity”?

In some familiar categories, we can think of the kernel $\ker f$ of an arrow $A\overset{f}{\rightarrow}B$ as a subset $\mathrm{Ker} f$ of its domain. In these cases, we usually think of the kernel as ...
11
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4answers
786 views

Group Theory via Category Theory

I have previously done a course on group theory and now I am doing a reading course on category theory. So as an interesting exercise I have been asked to write an exposition of group theory for ...
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2answers
150 views

Definition of codiagonal in a category

I'm confused by the definition of a codiagonal in a category with coproducts. The definition on nLab is as follows. Let $\mathcal{C}$ be a category with coproducts and let $X \in \mathcal{C}$. Then ...
4
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3answers
264 views

Colimits glue. What do limits do?

The informal but very useful way to think of colimits is as 'gluing things'. This intuitive view is very helpful to me, and I'd like one for limits aswell, but I haven't stumbled opon such a thing ...
2
votes
1answer
138 views

Tensor Product Construction, Solution Set Condition.

I am developing the basic properties of tensors, using categories. Let $R$ be a commutative ring. Fix $A, B\in R-Mod$ and define $K:R-Mod\rightarrow Set$ by $KC=\left \{ \beta :\left | A \right |\...
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1answer
52 views

Does a bijective homomorphism in Category theory have to be a bijection of arrows as well as objects?

In giving an example of two non-isomorphic posets related by a bijective homomorphism the following was proposed. $A=\{a,b,c\}$ with $a\leq b$, $a\leq c$ and $b$ and $c$ not comparable. $B=\{x,y,z\}$...
0
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1answer
48 views

Are Boo and BooRng really isomorphic?

In ACC on the top of page 34, I read: The construct Boo of boolean algebras is isomorphic to the construct BooRng of boolean rings and ring homomorphisms. This contradicts my intuition since ...
1
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1answer
123 views

Quotient objects as constructions from subobjects?

A quotient object of an object $A$ is usually denoted $A/B$ (we're talking about equivalence classes of epis). It seems that in categories like $\mathsf {Grp}$ and $\mathsf {Ab}$ one can associate ...
1
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1answer
147 views

An equivalence of categories that is not adjoint. [duplicate]

Is there a good example of an equivalence of categories $F:\mathcal{A}\to\mathcal{B}$ such that $F$ is neither left nor right adjoint to its inverse $G:\mathcal{B}\to\mathcal{A}$?
2
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1answer
87 views

A functor that has both left and right adjoints

What can we say about a functor that has both left and right adjoints? I vaguely recall hearing that it is then an equivalence of category. Is it true? If not, then under what conditions it is true?
2
votes
2answers
233 views

Construction of Yoneda extension

In "Category Theory" by Steve Awodey, there is a suggestion for the reader to construct a left adjoint in the proof of the UMP of the Yoneda embedding. Namely, for any small category $\mathbb{C}$, the ...
5
votes
1answer
81 views

Proving the 'letters' of a free group generate the group

A group $F$ is free over a set $X$ if there exists an injection $\sigma: X \to F$ such that for any function $\alpha: X \to G$ to any group $G$ there exists a unique homomorphism $\phi : F \to G$ such ...
7
votes
2answers
195 views

Is there an accepted term for those objects of a category $X$ such that for all $Y$, there is at most one arrow $X \rightarrow Y$?

In category theory, I have seen "weakly initial object" used as follows: $X$ is weakly initial iff for all objects $Y,$ there is at least one arrow $X \rightarrow Y$. Of course, another way of ...
1
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1answer
110 views

Computing $f^{*}\mathscr{O}_X$ directly via colimit

I want to prove that, for affine schemes $X = \text{Spec} (A)$, $Y = \text{Spec} (B)$ and $f: Y \rightarrow X$ morphism of schemes ($\varphi: A \rightarrow B$), $f^{*}\mathscr{O}_X \cong \mathscr{O}_Y$...
0
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1answer
83 views

Rel is a concrete category over Sets, but how to concretize that?

The traditional denotation of a structured set object is something like $(1)\qquad(X,\mathcal S_1,\dots\mathcal S_n)$ for some "structures" $\mathcal S_1,\dots\mathcal S_n$ on a set X. The modern ...
5
votes
1answer
144 views

Faithful functors from Rel, the category of sets and relations?

Are there examples of faithful functors $F:\mathbf{Rel}\to \mathbf C$, where C is a concrete category over Set? Or can it be proved that such functors don't exists?
7
votes
2answers
166 views

Why are there only limits and colimits?

Part of my intuition about the construction of limits and colimits is based on the idea that they are initial and terminal objects in the appropriate category: The limit of a diagram $D$ is of course ...
3
votes
1answer
84 views

“Any epi into a projective object clearly splits”

I am reading Category theory by Steve Awodey. So that we are all on the same page regarding the definitions I am using, I will repeat the definitions described in the book (starting on pg. 28): ...
2
votes
1answer
78 views

Difficulties with right-adjoint-right-inverse in $\mathsf{Top}$

I'm having difficulties with section $9$ of chapter $\mathrm V$ of CWM. There is the following proposition: Proposition 1. If $G:\mathsf C\rightarrow \mathsf D$ is a faithful functor, if $\mathsf ...
1
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1answer
90 views

Compact objects and locally finitely presentable categories (the Category of Groups)

I am trying to understand the concept of locally finitely presentable categories. I have discovered the concept of compact object here. I have discovered that for groups, the finitely presented ...
4
votes
2answers
245 views

What's the explicit categorical relation between a linear transformation and its matrix representation?

There several questions about linear transformations and its respective matrices in some basis, but I'm particularly interested in the explicit definition of this relation in the category $Vect$ (of ...
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2answers
76 views

inverse system vs inverse sequence

I am wondering about such problem. Let $\{X_i,\phi_{ij},I\}$ be an inverse system, where the directed set $I$ has such property that there exists a sequence $i_1 \leq i_2\leq\cdots\subset I$ such that ...
4
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1answer
75 views

nicer proof of basic functor category fact?

In functor categories, there's a nice isomorphism ${\mathcal C}^{\mathcal A \times \mathcal B} \cong ({\mathcal C}^{\mathcal B})^{\mathcal A}$. Proving this is a good exercise. It's not exactly hard, ...
1
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1answer
89 views

Intuition for the Discrete$\dashv$Forgetful$\dashv$Indiscrete Adjunction in $\mathsf{Top}$

Section 9 of CWM's chapter on limits beings by introducing the adjunctions $D\dashv U \dashv I$ where $D$ is the forgetful functor, $D$ equips sets with the discrete topology, and $I$ equips sets with ...
6
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1answer
115 views

Boolean algebra gives rise to a ring

There is a well-known result that every boolean algebra $(R,0,1,\land,\lor,\neg)$ can be made a boolean ring by defining $$x\cdot y=x\land y\qquad\text{and}\qquad x+y=(x\land\neg y)\lor(y\land\neg x)\,...
-1
votes
1answer
94 views

Common conditions on functions to be morphisms. [closed]

When coming in contact with the concept of morphism one may start to wonder what makes different structured objects of the same kind to be similar in a "morphical" way. At least I did. Below ...
1
vote
1answer
62 views

Explicitly describe colimits in $\mathsf{Set}$

I just started learning category theory a couple months ago. In my understanding, there is a nice fact about the category of sets that one can explicitly describe limits. If $F:J \to \mathsf{Set}$ is ...
2
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0answers
100 views

Categorical description of Alexander horned sphere?

Looking at the construction of the Alexander horned sphere: Is it possible to describe as some kind of direct limit of spaces? How can one formally see this?
4
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1answer
106 views

Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?

In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a ...
5
votes
1answer
102 views

sheaves of rings and maps to classifying topos

Let $R$ be the category of finitely presented commutative rings (but I don't know how necessary the hypothesis of finite presentation is for my question). Let $Set^R=Fun(R, Set)$ be the category of ...
5
votes
2answers
156 views

Categorical description of equivalence relation generated by a relation?

The notion of an equivalence relation generated by a relation is widespread and useful. How can one categorically describe it?
7
votes
2answers
1k views

Book for Algebraic Topology- Spanier vs Tom Dieck

A number of times, questions have been asked on this website about good books on Algebraic Topology and the responses have been very valuable. However I need some more specific advice in this matter. ...
2
votes
1answer
412 views

Universal object

I'm trying to define the concept of universal object in category theory, but I failed to find a universal way to do it. Maybe someone can help ? THe idea is quite simple. Let $\mathcal{C}$ be a ...
6
votes
1answer
164 views

Is there a universal mapping property satisfied by the disjoint union of totally ordered sets?

It is proved in another post that the product and coproduct do not exist in the category of totally ordered sets (except in some trivial cases). (In this post I will only consider the category TOrd, ...
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vote
0answers
89 views

Yoneda Embedding into Left Exact Functors

I think I am very confused about something. I've been reading a bit about the Mitchell embedding theorem, and I read that the proof first embeds a given small abelian category $\mathscr{A}$ into the ...
3
votes
1answer
145 views

Functors Between Functor Categories

I'm currently working through Tom Leinster's Basic Category Theory, and I have searched the internet fruitlessly for examples of functors between functor categories. I haven't yet come up with any ...
3
votes
0answers
70 views

Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
1
vote
1answer
70 views

Relating morphisms between two objects in category of finite sets and relations

In category of finite Sets and Relations (let's call it FinRel), suppose $A$ and $B$ are two objects (sets), and $b1,b2,b3$ are some morphisms (relations) from $A$ to $B$. Interpreting $b1 .. b3$ as ...
3
votes
0answers
65 views

Functors that preserve the subset relation on constructs

Is there a name for functors between concrete categories over Set that preserve '$\subset$'? Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ is a ...
0
votes
1answer
39 views

Biproduct in category

Let $(X\oplus X', \pi_X, \pi_{X'}, \iota_X,\iota_{X'})$ and $(Y \oplus Y', p_{Y'},p_Y, j_Y,j_{Y'})$ biproducts in a category $\mathcal{C}$. In MacLane's book, he defines using the structure of product,...
4
votes
2answers
120 views

Categorical Pasting Lemma

If I'm not mistaken, the pasting lemma for a two-element open cover $X=A\cup B$ of a topological space is equivalent to saying that the following square is a pushout in $\mathsf{Top}$: $$\begin{matrix}...
1
vote
1answer
31 views

Associativity of (co)Products

I have proved the following: Let $I=\bigcup _{k\in K}J_k$ be a partition of a set $I$. Consider of famility $(A_i)_{i\in I}$ of objects in a category $\mathsf C$. When all the products involved ...
3
votes
0answers
45 views

What are coquantales?

A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product. What are examples of non-diagonal comonoids in $\mathbf{Sup}...
4
votes
1answer
171 views

Categorical equivalents of Set theory concepts

Update: I am updating my question to be more precise. I am studding Category of finite sets and functions (FinSet). I am aware that for some of the concepts in Set Theory there are well-known ...
4
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0answers
80 views

Coequalizers are quotient maps

Is it possible to show that every coequalizer in the category of Hausdorff spaces is a quotient map directly from the universal property of a coequalizer and without use of the set-theoretical ...
0
votes
1answer
118 views

Query in the definion of abelian category

I am studying the definition of abelian category..Definition says it is a additive category with a)every morphism in category has kernel and co-kernel. b)every monomorphism in the category is the ...
0
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1answer
65 views

The $n$Lab on Currying

First of all, I have absolutely no knowledge in computer science. I am reading this in context with category theory, in particular the general tensor-hom adjunction. Suppose we're living in $\mathsf {...