6
votes
0answers
114 views

Continuous relations? [on hold]

What might it mean for a relation $R\subset X\times Y$ to be continuous? In topology, category theory or in analysis? Is it possible, canonical, useful? I have a vague idea of the possibility of ...
1
vote
0answers
202 views

Which are the constructs utilizing certain morphisms? [on hold]

It seems to be a fact that most mathematical constructs have canonical morphisms. In some cases, nevertheless, there is a choice between several different classes of morphisms. I found my way to ...
83
votes
9answers
4k 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 ...
6
votes
4answers
1k views

'Category-theory'-free areas of pure math, 'category-theory'-loaded areas of applied math

To put it short: In which active research areas of (pure) mathematics no (or only minimal) knowledge in category theory is required ? To put it long: I know almost nothing about category theory - but ...
5
votes
1answer
400 views

Is Logic/Set Theory necessary for studying Topos Theory?

I have just completed a postgraduate course, in which I studied Category Theory, without having a background in Set Theory and Logic - this probably already sounds absurd to many. This did not seem to ...
16
votes
3answers
1k views

Is there a scheme corresponding to the unit interval?

Can someone complete the following table? $\begin{array}{cc} \text{Topology over } \mathbb{R} & \text{Topology over } \mathbb{C} & \text{Algebraic Geometry} \\\\ \hline \mathbb{R} & ...
8
votes
6answers
1k views

Intuitionistic logic as quantization of classical logic?

A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with Intuitionistic logic. It is ...
43
votes
11answers
3k 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 ...
27
votes
3answers
1k views

“Softness” vs “rigidity” in Geometry

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ ...
5
votes
1answer
563 views

Convenient definition of “category of Riemannian manifolds”?

Has a notion of "category of Riemannian manifolds" been defined and used in the literature? For which reasons is it or would it (not) be a useful notion? I think the objects should be all (perhaps ...
12
votes
2answers
1k views

Examples of algorithms that came from category theory?

Generating Compiler Optimizations from Proofs is a wonderful paper. The authors say that they were faced with the problem, got stuck, then tried reasoning about it using category theory. They took ...
16
votes
2answers
1k views

Surveys of Goodwillie Calculus

Is there a good general introduction to Goodwillie calculus out there, like a paper or publication that gives a general overview of the calculus as well as how it is useful and why we are interested ...
33
votes
15answers
6k 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 ...
1
vote
0answers
271 views

Do Arbib and Manes describe just concrete categories?

In “Arbib, Manes. Arrows, Structures and Functors. The Categorical Imperative. 6. Structured sets.” there is an approach to formalize structures. I have a strong feeling that they describe just ...
21
votes
3answers
2k views

Lawvere theories versus classical universal algebra

A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with ...
6
votes
3answers
282 views

2-morphisms in structured 2-categories

There are many $2$-categories, which are first specified by certain categories with extra structure; then the $1$- and $2$-morphisms are functors and natural transformations that preserve the extra ...
19
votes
3answers
2k views

Locales and Topology.

As someone more used to point-set topology, who is unfamiliar with the inner workings of lattice theory, I am looking to learn about the localic interpretation of topology, of which I only have a ...
7
votes
0answers
664 views

triangulated/derived categories in Physics and algebraic geometry

Why do physicists care about the triangulated/derived categories? I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...
1
vote
1answer
1k views

Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...
12
votes
1answer
670 views

evil properties, higher category theory and well-chosen tensor products

Let's start with the following random example: If $F$ is a presheaf, then for every chain of open subsets $U \subseteq V \subseteq W$, the morphisms $F(W) \to F(V) \to F(U)$ and $F(W) \to F(U)$ ...
0
votes
1answer
134 views

Isomorphism by classification

Are there any examples other than using dimension for vector spaces where the easiest way to show that two objects are isomorphic is by using a classification theorem and showing that they must both ...
29
votes
5answers
3k views

Why do categorical foundationalists want to escape set theory?

This is a question that I have seen asked passively in comments relating to the separation of category theory from set theory, but I haven't seen it addressed in full. I know that it's possible to ...
10
votes
6answers
2k views

What do people mean by “subcategory”?

Mac Lane defines a subcategory as a subset of objects and a subset of morphisms that form a category. But the first rule of category theory is that you do not talk about equality of objects. Up to ...
32
votes
13answers
2k views

Equality vs. isomorphism vs. specific isomorphism

This question prompted a reformulation: What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit? I believe this to be a serious question because ...
27
votes
4answers
3k views

Is “all categorical reasoning formally contradictory”?

In the December 2009 issue of the newsletter of the European Mathematical Society there is a very interesting interview with Pierre Cartier. In page 33, to the question What was the ontological ...
23
votes
8answers
3k views

Geometric intuition for limits

I'm the sort of mathematician who works really well with elements. I really enjoy point-set topology, and category theory tends to drive me crazy. When I was given a bunch of exercises on subjects ...
5
votes
4answers
2k views

Why are inverse images more important than images in mathematics?

Why are inverse images of functions more central to mathematics than the image? I have a sequence of related questions: Why the fixation on continuous maps as opposed to open maps? (Is there an ...
8
votes
6answers
3k views

why haven't certain well-researched classes of mathematical object been framed by category theory?

Category theory is doing/has done a stellar job on Set, FinSet, Grp, Cob, Vect, cartesian closed categories provide a setting for $\lambda$-calculus, and Baez wrote a paper (Physics, Topology, Logic ...
8
votes
3answers
772 views

What's the “correct” smooth structure on the category of manifolds?

As will become clear, this is in some sense a follow up on my earlier question Why should I prefer bundles to (surjective) submersions?. As with that one, I hope that it's not too open-ended or ...
40
votes
11answers
10k 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: Joyals ...
11
votes
3answers
1k views

Applications of homotopy groups of spheres

The study of the homotopy groups of spheres $\pi_i(S^n)$ is a major subject in algebraic topology. One knows for example that nearly all of them are finite groups. Some are explicitly known. There is ...
29
votes
7answers
3k views

What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
5
votes
4answers
908 views

Why is the concept of topos a “metamorphosis” of the concept of space?

Hi, I recently started studying topos theory, and I am puzzled by the Grothendieck's claim that topos is a "metamorphosis" of the concept of space. Can somebody explain what he means by this? ...
12
votes
4answers
2k views

Category theory and model theory as “natural” counterparts

I am aware of the profound discussion of the relationship between category theory and model theory (e.g. at The n-Category Café) but do wonder why category theory (CT) is not opposed to model theory ...
10
votes
2answers
409 views

Primacy of arcs/arrows over vertices/objects

Freyd's Abelian Categories is the only textbook I know where the primacy of arrows over objects is taken seriously already in the axioms: there is no talk of objects at all. Only later one sees, that ...
1
vote
6answers
475 views

Between abstract and concrete: What's the right way to think of specific categories?

At the risk of annoying some of the categorists I feel urged to pose this beginner-ish question: If one talks about a specific category such as the category of sets with functions or the category of ...
14
votes
11answers
2k views

Learning to Think Categorically

Up to this point in my education, I have had very little exposure to the language and machinery of category theory, and I would like to rectify this. My goal is to become conversant with some of the ...