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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 preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

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In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
Ben Blum-Smith's user avatar
45 votes
1 answer
4k views

Equivalent conditions for a preabelian category to be abelian

Let's fix some terminology first. A category $\mathcal{C}$ is preabelian if: 1) $Hom_{\mathcal{C}}(A,B)$ is an abelian group for every $A,B$ such that composition is biadditive, 2) $\mathcal{C}$ has ...
Bruno Stonek's user avatar
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271 votes
6 answers
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In (relatively) simple words: What is an inverse limit?

I am a set theorist in my orientation, and while I did take a few courses that brushed upon categorical and algebraic constructions, one has always eluded me. The inverse limit. I tried to ask one of ...
Asaf Karagila's user avatar
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37 votes
1 answer
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Analogue of the Cantor-Bernstein-Schroeder theorem for general algebraic structures

The Cantor-Bernstein-Schroeder theorem states that if there are two sets $A$ and $B$ such that there exist injective (alternatively, surjective, assuming choice I think) maps $A \to B$ and $B \to A$, ...
Carl's user avatar
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24 votes
5 answers
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Abelianization of free group is the free abelian group

How does one prove that if $X$ is a set, then the abelianization of the free group $FX$ on $X$ is the free abelian group on $X$?
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237 votes
27 answers
77k views

Good books and lecture notes about category theory.

What are the best books and lecture notes on category theory?
61 votes
2 answers
7k views

Category-theoretic limit related to topological limit?

Is there any connection between category-theoretic term 'limit' (=universal cone) over diagram, and topological term 'limit point' of a sequence, function, net...? To be more precise, is there a ...
Rafael Mrden's user avatar
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37 votes
3 answers
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The "magic diagram" is cartesian

I am trying to solve an exercise from Vakil's lecture notes on algebraic geometry, namely, I want to show that $\require{AMScd}$ \begin{CD} X_1\times_Y X_2 @>>> X_1\times_Z X_2\\ @V V ...
Paul's user avatar
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33 votes
1 answer
22k views

Hom is a left-exact functor

If $$ 0 \to A \to B\to C,$$ is a left exact sequence of $R$-module, then for any $R$-module $M$, $$ 0 \to \operatorname{Hom}_R(M,A)\to \operatorname{Hom}_R(M,B)\to \operatorname{Hom}_R(M,C), $...
Gobi's user avatar
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21 votes
6 answers
3k views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
Shaun's user avatar
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13 votes
4 answers
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Why doesn't the definition of a category require an explicit notion of morphisms equality?

I am learning basics of the category theory (CT). I do understand that CT is a modern powerful framework to describe various branches of mathematics in a unified way. I do admit that category's ...
Zazaeil's user avatar
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96 votes
9 answers
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What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the ...
Hans-Peter Stricker's user avatar
54 votes
2 answers
7k views

Yoneda-Lemma as generalization of Cayley`s theorem?

I came across the statement that Yoneda-lemma is a generalization of Cayley`s theorem which states, that every group is isomorphic to a group of permutations. How exactly is Yoneda-lemma a ...
Jan's user avatar
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47 votes
4 answers
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Proving the snake lemma without a diagram chase

Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A' \mathrel{\overset{f'}{\to}} B' \...
Zhen Lin's user avatar
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39 votes
2 answers
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Can it happen that the image of a functor is not a category?

On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a ...
Anonymous Coward's user avatar
39 votes
6 answers
5k views

What's the deal with empty models in first-order logic?

Asaf's answer here reminded me of something that should have been bothering me ever since I learned about it, but which I had more or less forgotten about. In first-order logic, there is a convention ...
Qiaochu Yuan's user avatar
22 votes
1 answer
1k views

Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?

In this question, The Chaz asks whether $G\times G\cong H\times H$ implies that $G\cong H$, where $G$ and $H$ are finite abelian groups. The answer is to his question is yes, by the structure theorem ...
Myself's user avatar
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18 votes
1 answer
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universal property in quotient topology

The following is a theorem in topology: Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. Let $\pi: X\to X/\sim$ be the canonical projection. If $g : X → Z$ is a continuous ...
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16 votes
3 answers
2k views

Do opposite categories always exist?

The opposite category of C is defined by setting hom(x,y)=hom(y,x) for all objects x,y of C. However in a concrete category, morphisms do not always have inverses (because functions don't always have ...
edenstar's user avatar
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7 votes
2 answers
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Let $C,D$ be categories and $F:C\to D$ and $G:D\to C$ be adjoint functors. Then $F$ is fully faithful iff the unit is an isomorphism?

Let $C,D$ be categories and $F:C\to D,G:D\to C$ be such that $F$ is a left adjoint of $G$. Prove that $F$ is fully faithful iff the unit is an isomorphism. (This is an exercise from the book by T. ...
PLE's user avatar
  • 391
146 votes
4 answers
34k views

What is category theory useful for?

Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says it ...
Asinomás's user avatar
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31 votes
3 answers
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Elements in $\hat{\mathbb{Z}}$, the profinite completion of the integers

Let $\hat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$. Since $\hat{\mathbb{Z}}$ is the inverse limit of the rings $\mathbb{Z}/n\mathbb{Z}$, it's a subgroup of $\prod_n \mathbb{Z}/n\mathbb{...
Mike Battaglia's user avatar
23 votes
2 answers
2k views

What is the intuition behind short exact sequences of groups; in particular, what is the intuition behind group extensions?

What is the intuition behind short exact sequences of groups; in particular, what is the intuition behind group extensions? I'm sorry that the definitions below are a bit haphazard but they're how I ...
Shaun's user avatar
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18 votes
4 answers
1k views

Construction of a Hausdorff space from a topological space

Let $X$ be a topological space. Is there a Hausdorff space $HX$ and a continuous function $i:X\rightarrow HX$ such that for any Hausdorff space $A$ and a continuous function $j:X\rightarrow A$, there ...
Amr's user avatar
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128 votes
8 answers
30k views

When to learn category theory?

I'm a undergraduate who wishes to learn category theory but I only have basic knowledge of linear algebra and set theory, I've also had a short course on number theory which used some basic concepts ...
Vicfred's user avatar
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23 votes
1 answer
5k views

Is an equivalence an adjunction?

Let $C$ and $D$ be categories and $F:C\to D$, $G:D\to C$ two functors. $F$ is left-adjoint to $G$, if there are natural transformations $\eta:id_C\to GF$ and $\epsilon:FG\to id_D$ such that \begin{...
sopot's user avatar
  • 555
10 votes
3 answers
6k views

Group as a category

Is it possible to define a group as a category? What exactly will be objects of this category and how will we say that every element should have an inverse?
Mohan's user avatar
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8 votes
4 answers
1k views

Terminal objects as "nullary" products

I read something weird in my category theory book (Awodey p 47). " Observe also that a terminal object is a nullary product, that is, a product of no objects: Given no objects, there is an object $1$...
Vinyl_cape_jawa's user avatar
76 votes
5 answers
7k views

Why should I care about adjoint functors

I am comfortable with the definition of adjoint functors. I have done a few exercises proving that certain pairs of functors are adjoint (tensor and hom, sheafification and forgetful, direct image and ...
DBr's user avatar
  • 4,790
47 votes
1 answer
2k views

Functions $f:\mathbb{N}\rightarrow \mathbb{Z}$ such that $\left(m-n\right) \mid \left(f(m)-f(n)\right)$

A long time back, I wondered what functions other than integer polynomials on $\mathbb{N}$ (or $\mathbb{Z}$) satisfied the property: $$\forall m,n: \left(m-n\right) \mid \left(f(m)-f(n)\right)$$ ...
42 votes
3 answers
5k views

How to define Homology Functor in an arbitrary Abelian Category?

In the Category of Modules over a Ring, the i-th Homology of a Chain Complex is defined as the Quotient Ker d / Im d where d as usual denotes the differentials, indexes skipped for simplicity. How ...
user avatar
33 votes
3 answers
4k views

Why is every category not isomorphic to its opposite?

This is a beginner category theory question: I'm trying to wrap my head around the fact that we do not have $\mathbf{Sets} \cong \mathbf{Sets}^\mathsf{op}$, i.e., the category of sets is not ...
Ulrik Rasmussen's user avatar
33 votes
7 answers
2k views

Why are continuous functions the "right" morphisms between topological spaces?

Recently, someone mentioned to me that given a function $f: X \to Y$ there are two natural functions between the powersets $P(X)$ and $P(Y)$. Namely $f: U \subset X \mapsto f(U)$ and $f^{-1}: V \...
Alexander's user avatar
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33 votes
4 answers
6k views

Reference request: compact objects in R-Mod are precisely the finitely-presented modules?

Let $R$ be a ring. According to this MO question, the modules $M \in R\text{-Mod}$ such that $\text{Hom}(M, -)$ preserves all filtered colimits (the compact objects) are precisely the finitely-...
Qiaochu Yuan's user avatar
19 votes
5 answers
7k views

Examples of categories where epimorphism does not have a right inverse, not surjective

An epimorphism is defined as follows: $f \in \operatorname{Hom}_C(A,B)$ is an epimorphism if $\forall Z, \forall h', h'' \in \operatorname{Hom}_C(B, Z)$ then $h' f = h'' f \; \Rightarrow \; h' = h''...
Daniil's user avatar
  • 1,167
10 votes
3 answers
2k views

Exponential objects in a cartesian closed category: $a^1 \cong a$

Hi I'm having problems with coming up with a proof for this simple property of cartesian closed categories (CCC) and exponential objects, namely that for any object $a$ in a CCC $C$ with an initial ...
Anas's user avatar
  • 145
10 votes
1 answer
2k views

Does this "extension property" for polynomial rings satisfy a universal property?

On page 151 of Paolo Aluffi's Algebra: Chapter 0, an important property of the polynomial ring $\mathbb{Z}[x_1, \cdots, x_n]$ is introduced, namely that it's initial in the category of set functions ...
Alf's user avatar
  • 2,597
7 votes
2 answers
2k views

Adjoint functors

I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor ...
Tom Crockett's user avatar
5 votes
2 answers
272 views

Is this an equivalent formulation of "surjective" resp. "epimorphism"?

An $A$-element $x$ of $X$, written $x\in_A X$, is a map $x:A\to X$. If $f$ is a map with domain $X$, and $x\in_A X$ is an element, we write $f(x)$ to denote the composite of $f$ and $x$. Now say that ...
user avatar
4 votes
3 answers
843 views

Group theoretic meaning of natural isomorphisms between certain functors

So imagine you have a group $G$ and we consider the set of group homomorphisms from $\mathbb{Z}$ to $G$ specified by $\forall g$ $\in G$ $\exists$ $\phi(1)=g$. Each of these homomorphisms is in 1-...
East's user avatar
  • 351
2 votes
1 answer
179 views

Examples of conjugate-like structures across mathematics

This is a flavor question rather than a specific problem. I'm an undergrad, and I've noticed a common tactic used to understand and break down mathematical objects, which looks a lot like conjugacy in ...
While I Am's user avatar
  • 2,454
71 votes
8 answers
4k views

A bestiary about adjunctions

What is your favourite adjoint? Following Mac Lane philosophy adjoints are everywhere, so I would like to draw a (possibly but unprobably) exhaustive list of adjunctions one faces in studying ...
56 votes
7 answers
6k views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed "...
Amr's user avatar
  • 20.1k
36 votes
3 answers
9k views

Is Category Theory similar to Graph Theory?

The following author noted: Roughly speaking, category theory is graph theory with additional structure to represent composition. My question is: Is Category Theory similar to Graph Theory?
hawkeye's user avatar
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35 votes
3 answers
14k views

Epimorphism and Monomorphism = Isomorphism?

It seems to be that if a map is both an epimorphism and a monomorphism, it is not necessarily the case that it is an isomorphism. However, in the category of sets, if a map is both an epimorphism and ...
Gerber's user avatar
  • 867
31 votes
5 answers
3k views

Maps in Opposite Categories

Given some category ${\mathcal C}$, then the opposite category will consist of the same objects with the morphisms "turned around." Given $f:A\rightarrow B$, for $A,B$ objects of ${\mathcal C}$, then ...
user avatar
30 votes
6 answers
8k views

Right adjoints preserve limits

In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction,...
Bruno Stonek's user avatar
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28 votes
3 answers
6k views

Why is there no functor $\mathsf{Group}\to\mathsf{AbGroup}$ sending groups to their centers?

The category $\mathbf{Set}$ contains as its objects all small sets and arrows all functions between them. A set is "small" if it belongs to a larger set $U$, the universe. Let $\mathbf{Grp}$ be ...
Corey545's user avatar
  • 289
25 votes
3 answers
3k views

What are the epimorphisms in the category of Hausdorff spaces?

It appears to be the case that the epimorphisms in $\text{Haus}$ are precisely the maps with dense image. This is claimed in various places, but a comment on my blog has made me doubt the source I got ...
Qiaochu Yuan's user avatar
19 votes
1 answer
6k views

Proof that the tensor product is the coproduct in the category of R-algebras

Given the category of commutative R- or k-Algebras, it is often mentioned that the coproduct is the same as the tensor product. I'm interested in the proof of this statement. One idea would be to ...
cbb's user avatar
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