Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
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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 ...
150
votes
12
answers
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"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
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10
answers
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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
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11
answers
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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 ...
106
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15
answers
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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
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7
answers
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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 ...
93
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9
answers
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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
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8
answers
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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
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10
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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. ...
90
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10
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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, ...
87
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5
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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
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4
answers
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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 ...
86
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1
answer
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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 ...
81
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9
answers
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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 ...
79
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5
answers
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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 ...
78
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9
answers
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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
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6
answers
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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,...
75
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13
answers
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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.
75
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8
answers
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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 ...
75
votes
5
answers
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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
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6
answers
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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. ...
73
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13
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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
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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 ...
72
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8
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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 ...
72
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6
answers
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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? ...
71
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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
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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 ...
71
votes
3
answers
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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 ...
71
votes
1
answer
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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 ...
70
votes
28
answers
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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
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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
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2
answers
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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 ...
66
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4
answers
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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 ...
66
votes
4
answers
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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
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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 ...
64
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6
answers
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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 ...
64
votes
2
answers
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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
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22
answers
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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
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5
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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 ...
62
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4
answers
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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 ...
61
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13
answers
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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
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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 ...
61
votes
3
answers
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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?...
60
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8
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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 ...
60
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6
answers
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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 ...
59
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8
answers
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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 ...
56
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3
answers
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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. ...
56
votes
1
answer
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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 ...
55
votes
3
answers
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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 ...
55
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6
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(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 ...