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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 (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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243 votes
1 answer
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Is this really a categorical approach to integration?

Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A ...
Shaun's user avatar
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236 votes
27 answers
77k views

Good books and lecture notes about category theory.

What are the best books and lecture notes on category theory?
172 votes
1 answer
3k views

Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
user avatar
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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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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113 votes
6 answers
15k views

Why don't analysts do category theory?

I'm a mathematics student in abstract algebra and algebraic geometry. Most of my books cover a great deal of category theory and it is an essential tool in understanding these two subjects. Recently, ...
gary's user avatar
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104 votes
1 answer
5k views

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
96 votes
9 answers
25k views

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
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
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73 votes
5 answers
12k views

Can someone explain the Yoneda Lemma to an applied mathematician?

I have trouble following the category-theoretic statement and proof of the Yoneda Lemma. Indeed, I followed a category theory course for 4-5 lectures (several years ago now) and felt like I understood ...
Chris Taylor's user avatar
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 ...
67 votes
6 answers
10k views

Is it possible to formulate category theory without set theory?

I have never understood why set theory has so many detractors, or what is gained by avoiding its use. It is well known that the naive concept of a set as a collection of objects leads to logical ...
Matt Calhoun's user avatar
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66 votes
1 answer
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Abstract nonsense proof of snake lemma

During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
Jesko Hüttenhain's user avatar
64 votes
3 answers
12k views

What use is the Yoneda lemma?

Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse ...
Alex Becker's user avatar
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63 votes
3 answers
13k views

Why is it worth spending time on type theory?

Looking around there are three candidates for "foundations of mathematics": set theory category theory type theory There is a seminal paper relating these three topics: From Sets to Types to ...
Hans-Peter Stricker's user avatar
61 votes
2 answers
4k views

What are exact sequences, metaphysically speaking?

Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
Joshua Seaton's user avatar
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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60 votes
2 answers
9k views

Why are (representations of ) quivers such a big deal?

Quivers are directed graphs where loops and multi-arrows are allowed. And we can talk about representations of quivers by assigning each vertex a vector space and each arrow a homomorphism. Moreover, ...
Hui Yu's user avatar
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60 votes
3 answers
3k views

Is Stokes' Theorem natural in the sense of category theory?

Stokes' Theorem asserts that for a compactly-supported differential form $\omega$ of degree $n-1$ on a smooth oriented $n$-dimensional manifold $M$ we have the marvellous equation $$\int_M d\omega = \...
Martin Brandenburg's user avatar
58 votes
8 answers
3k views

What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other words,...
Qiaochu Yuan's user avatar
57 votes
5 answers
13k views

Category of all categories vs. Set of all sets

In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...
Nick Alger's user avatar
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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
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56 votes
3 answers
9k views

Cokernels - how to explain or get a good intuition of what they are or might be

When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over ...
Mark Bennet's user avatar
56 votes
1 answer
7k views

Textbooks on higher category theory

What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv ...
Martin Brandenburg'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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52 votes
9 answers
25k views

Real world applications of category theory

I was reading some basic information from Wiki about category theory and honestly speaking I have a very weak knowledge about it. As it sounds interesting, I will go into the theory to learn more if ...
Seyhmus Güngören's user avatar
52 votes
3 answers
13k views

Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...
Casebash's user avatar
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51 votes
8 answers
6k views

Category-theoretic description of the real numbers

The familiar number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ all have "natural constructions", which indicate, why they are mathematically interesting. For example, equipping $\mathbb{N}$ with ...
user avatar
51 votes
1 answer
3k views

Does "cheap nonstandard analysis" take place in a topos?

Terence Tao's A cheap version of nonstandard analysis describes a way to do analysis halfway between ordinary analysis and nonstandard analysis which, if I'm not mistaken, cashes out to working in the ...
Qiaochu Yuan's user avatar
49 votes
4 answers
6k views

Natural and coordinate free definition for the Riemannian volume form?

In linear algebra and differential geometry, there are various structures which we calculate with in a basis or local coordinates, but which we would like to have a meaning which is basis independent ...
ziggurism's user avatar
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47 votes
1 answer
4k views

Why is the category of fields seemingly so poorly behaved?

Compared to the categories of other “common” algebraic objects like groups and rings, it seems that fields as a whole are missing some important properties: There are no initial or terminal objects ...
Ducky's user avatar
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47 votes
4 answers
9k views

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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47 votes
1 answer
6k views

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
t.b.'s user avatar
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47 votes
1 answer
7k views

Quotient objects, their universal property and the isomorphism theorems

This is a question that has been bothering me for quite a while. Let me put between quotation marks the terms that are used informally. "Quotient objects" are always the same. Take groups, abelian ...
Bruno Stonek's user avatar
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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: $$\forall m,n: \left(m-n\right) \mid \left(f(m)-f(n)\right)$$ ...
46 votes
1 answer
10k 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 ...
Asinomás's user avatar
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45 votes
9 answers
7k views

Mnemonic for the fact that a right(left) adjoint functor preserves limits(colimits)

A right adjoint functor preserves limits. Dually a left adjoint functor preserves colimits. I often forget which is which. Of course, you can look up a book on category theory or use internet. But it'...
Makoto Kato's user avatar
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45 votes
4 answers
2k views

What does a proof in an internal logic actually look like?

The nLab has a lot of nice things to say about how you can use the internal logic of various kinds of categories to prove interesting statements using more or less ordinary mathematical reasoning. ...
Qiaochu Yuan'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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44 votes
4 answers
3k views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
Ben Blum-Smith's user avatar
43 votes
8 answers
4k views

Why are algebraic structures preserved under intersection but not union?

In general, the intersection of subgroups/subrings/subfields/sub(vector)spaces will still be subgroups/subrings/subfields/sub(vector)spaces. However, the union will (generally) not be. Is there a "...
MathematicsStudent1122's user avatar
43 votes
4 answers
4k views

Are diffeomorphic smooth manifolds truly equivalent?

It seems to be an often repeated, "folklore-ish" statement, that diffeomorphism is an equivalence relation on smooth manifolds, and two smooth manifolds that are diffeomorphic are indistinguishable in ...
Bence Racskó's user avatar
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
40 votes
2 answers
3k views

Category Theory and Lebesgue Integration.

I'm wondering if there's any Category Theory floating around in the theory of Lebesgue Integration. To avoid things becoming too broad, let's keep this focused on the basics. Here's how I see the ...
Shaun's user avatar
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40 votes
1 answer
7k views

What are some of the major open problems in category theory?

What are some of the major open problems in category theory? Just curious - I'm interested in category theory.
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
39 votes
2 answers
4k views

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
4 answers
2k views

Why do we look at morphisms?

I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure (...
readingframe's user avatar
39 votes
2 answers
5k views

Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
Arrow's user avatar
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