Questions tagged [ct.category-theory]

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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230 votes
13 answers
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Is there an introduction to probability theory from a structuralist/categorical perspective?

The title really is the question, but allow me to explain. I am a pure mathematician working outside of probability theory, but the concepts and techniques of probability theory (in the sense of ...
Pete L. Clark's user avatar
150 votes
12 answers
41k views

"Philosophical" meaning of the Yoneda Lemma

The Yoneda Lemma is a simple result of category theory, and its proof is very straightforward. Yet I feel like I do not truly understand what it is about; I have seen a few comments here mentioning ...
146 votes
10 answers
15k views

What non-categorical applications are there of homotopical algebra?

(To be honest, I actually mean something more general than 'homotopical algebra' - topos theory, $\infty$-categories, operads, anything that sounds like its natural home would be on the nLab.) More ...
108 votes
11 answers
12k views

Why do Groups and Abelian Groups feel so different?

Groups are naturally "the symmetries of an object". To me, the group axioms are just a way of codifying what the symmetries of an object can be so we can study it abstractly. However, this heuristic ...
Greg Muller's user avatar
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106 votes
15 answers
35k views

Most striking applications of category theory?

What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples: Joyal's ...
94 votes
7 answers
9k views

When does Cantor-Bernstein hold?

The Cantor-Bernstein theorem in the category of sets (A injects in B, B injects in A => A, B equivalent) holds in other categories such as vector spaces, compact metric spaces, Noetherian topological ...
Randomblue's user avatar
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93 votes
9 answers
36k views

Is Mac Lane still the best place to learn category theory?

For a student embarking on a study of algebraic topology, requiring a knowledge of basic category theory, with a long-term view toward higher/stable/derived category theory, ... Is Mac Lane still ...
91 votes
8 answers
15k views

Has incorrect notation ever led to a mistaken proof?

In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...
91 votes
10 answers
14k views

Reflection principle vs universes

In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...
Peter Scholze's user avatar
90 votes
10 answers
17k views

How do I check if a functor has a (left/right) adjoint?

Because adjoint functors are just cool, and knowing that a pair of functors is an adjoint pair gives you a bunch of information from generalized abstract nonsense, I often find myself saying, "Hey, ...
Harrison Brown's user avatar
87 votes
5 answers
15k views

Why higher category theory?

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
87 votes
4 answers
7k views

Is there an accepted definition of $(\infty,\infty)$ category?

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with ...
Theo Johnson-Freyd's user avatar
86 votes
1 answer
5k views

Are there non-scalar endomorphisms of the functor $V\mapsto V^{**}/V$?

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor $$ V\mapsto V^{**}/V $$ of the category of $K$-vector spaces? I asked a related question on Mathematics Stackexchange, but ...
Pierre-Yves Gaillard's user avatar
81 votes
9 answers
8k views

What are some examples of interesting uses of the theory of combinatorial species?

This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by this thread, and sigfpe's comment to Pete Clark's answer. I've often heard it claimed ...
Pietro's user avatar
  • 1,570
79 votes
5 answers
5k views

Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

The Lawvere fixed point theorem asserts that if $X, Y$ are objects in a category with finite products such that the exponential $Y^X$ exists, and if $f : X \to Y^X$ is a morphism which is surjective ...
Qiaochu Yuan's user avatar
78 votes
9 answers
21k views

Results that are widely accepted but no proof has appeared

The background of this question is the talk given by Kevin Buzzard. I could not find the slides of that talk. The slides of another talk given by Kevin Buzzard along the same theme are available here. ...
78 votes
6 answers
6k views

Rigidity of the category of schemes

Call a category $C$ rigid if every equivalence $C \to C$ is isomorphic to the identity. I don't know if this is standard terminology. Many of the usual algebraic categories are rigid, for example sets,...
Martin Brandenburg's user avatar
75 votes
13 answers
12k views

What precisely Is "Categorification"?

(And what's it good for.) Related MO questions (with some very nice answers): examples-of-categorification; can-we-categorify-the-equation $(1-t)(1+t+t^2+\dots)=1$?; categorification-request.
Gil Kalai's user avatar
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75 votes
8 answers
17k views

What is an intuitive view of adjoints? (version 1: category theory)

In trying to think of an intuitive answer to a question on adjoints, I realised that I didn't have a nice conceptual understanding of what an adjoint pair actually is. I know the definition (several ...
Andrew Stacey's user avatar
75 votes
5 answers
3k views

When the automorphism group of an object determines the object

Let me start with three examples to illustrate my question (probably vague; I apologize in advance). $\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...
73 votes
6 answers
5k views

whence commutative diagrams?

It seems that commutative diagrams appeared sometime in the late 1940s -- for example, Eilenberg-McLane (1943) group cohomology paper does not have any, while the 1953 Hochschild-Serre paper does. ...
Igor Rivin's user avatar
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73 votes
13 answers
6k views

Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations. Why was there the necessity of singling ...
73 votes
1 answer
8k views

Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories. Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious ...
Steven Landsburg's user avatar
72 votes
8 answers
12k views

Category theory and set theory: just a different language, or different foundation of mathematics?

This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics. I am asking for a reference. In order to make the reference request as ...
Claus's user avatar
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72 votes
6 answers
6k views

A bestiary of topologies on Sch

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? ...
David Roberts's user avatar
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71 votes
16 answers
20k views

Is there a nice application of category theory to functional/complex/harmonic analysis?

[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.] I've read looked at the examples in most ...
71 votes
10 answers
19k views

Relating category theory to programming language theory

I'm wondering what the relation of category theory to programming language theory is. I've been reading some books on category theory and topos theory, but if someone happens to know what the ...
Michael Hoffman's user avatar
71 votes
3 answers
17k views

What is Koszul duality?

Okay, let's make sure I'm on the same page with those who know homological algebra. What is Koszul duality in general? What does it mean that categories are Koszul dual (I guess representations of ...
Ilya Nikokoshev's user avatar
71 votes
1 answer
2k views

Dualizing the notion of topological space

$\require{AMScd}$ Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements ...
Ronald J. Zallman's user avatar
70 votes
28 answers
7k views

Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon: (1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...
69 votes
7 answers
14k views

The main theorems of category theory and their applications

This question first arose as I wrote an answer for the question: Is there a nice application of category theory to functional/complex/harmonic analysis?; it can also be regarded as a (hopefully) more ...
67 votes
2 answers
14k views

Is there a category structure one can place on measure spaces so that category-theoretic products exist?

The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...
Damek Davis's user avatar
66 votes
4 answers
5k views

Why did Voevodsky consider categories "posets in the next dimension", and groupoids the correct generalisation of sets?

Earlier today, I stumbled upon this article written by V. Voevodsky about the "philosophy" behind the Univalent Foundations program. I had read it before around the time of his passing, and one ...
Soham Chowdhury's user avatar
66 votes
4 answers
11k views

What's there to do in category theory?

Disclaimer: I posted this question on MSE only a few days ago; and received very few comments. I know that the etiquette is to wait a bit more than that before moving a post from MSE to MO, but I ...
66 votes
3 answers
4k views

Does linearization of categories reflect isomorphism?

Given a category $C$ and a commutative ring $R$, denote by $RC$ the $R$-linearization: this is the category enriched over $R$-modules which has the same objects as $C$, but the morphism module between ...
Tilman's user avatar
  • 6,032
64 votes
6 answers
9k views

Are dagger categories truly evil?

Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...
André Henriques's user avatar
64 votes
2 answers
13k views

What is descent theory?

I read the article in wikipedia, but I didn't find it totally illuminating. As far as I've understood, essentially you have a morphism (in some probably geometrical category) $Y \rightarrow X$, where ...
62 votes
22 answers
18k views

What's a groupoid? What's a good example of a groupoid? [closed]

Or more specifically, why do people get so excited about them? And what's your favorite easy example of one, which illustrates why I should care (and is not a group)?
62 votes
5 answers
9k views

Does "finitely presented" mean "always finitely presented"? (Answered: Yes!)

Precisely, if an R-module M has a finite presentation, and Rk → M is some unrelated surjection (k finite), is the kernel necessarily also finitely generated? Basically I want to believe I can ...
Andrew Critch's user avatar
62 votes
4 answers
6k views

When size matters in category theory for the working mathematician

I think a related question might be this (Set-Theoretic Issues/Categories). There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance ...
jg1896's user avatar
  • 2,683
61 votes
13 answers
8k views

Example of an unnatural isomorphism

Can anyone give an example of an unnatural isomorphism? Or, maybe, somebody can explain why unnatural isomorphisms do not exist. Consider two functors $F,G: {\mathcal C} \rightarrow {\mathcal D}$. We ...
61 votes
3 answers
6k views

Why is there no Cayley's Theorem for rings?

Cayley's theorem makes groups nice: a closed set of bijections is a group and a group is a closed set of bijections- beautiful, natural and understandable canonically as symmetry. It is not so much a ...
Tom Boardman's user avatar
  • 3,190
61 votes
3 answers
8k views

Why do filtered colimits commute with finite limits?

It's not hard to show that this is true in the category Set, and proofs have been written down in many places. But all the ones I know are a bit fiddly. Question 1: is there a soft proof of this fact?...
Steve Lack's user avatar
  • 3,081
60 votes
8 answers
6k views

Is the ultraproduct concept fundamentally category-theoretic?

Once again, I would like to take advantage of the large number of knowledgable category theorists on this site for a question I have about category-theoretic aspects of a fundamental logic concept. My ...
Joel David Hamkins's user avatar
60 votes
6 answers
10k views

Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
ಠ_ಠ's user avatar
  • 5,933
59 votes
8 answers
7k views

Natural transformations as categorical homotopies

Every text book I've ever read about Category Theory gives the definition of natural transformation as a collection of morphisms which make the well known diagrams commute. There is another possible ...
Giorgio Mossa's user avatar
56 votes
3 answers
9k views

What are the benefits of viewing a sheaf from the "espace étalé" perspective?

I learned the definition of a sheaf from Hartshorne—that is, as a (co-)functor from the category of open sets of a topological space (with morphisms given by inclusions) to, say, the category of sets. ...
Simon Rose's user avatar
  • 6,240
56 votes
1 answer
2k views

History: What was the Lemma? (Grothendieck Harvard Lectures; Mumford)

In an article about the life of Grothendieck, available here: http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf Allyn Jackson writes about how Mumford was profoundly impressed: Mumford ...
Dikran Karagueuzian's user avatar
55 votes
3 answers
3k views

Duality between compactness and Hausdorffness

Consider a non-empty set $X$ and its complete lattice of topologies (see also this thread). The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...
yada's user avatar
  • 1,731
55 votes
6 answers
15k views

(How) is category theory actually useful in actual physics?

An answer to a recent question motivated the following question: (how) is category theory actually useful in actual physics? By "actual physics" I mean to refer to areas where the underlying ...
Steve Huntsman's user avatar

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