# Tagged Questions

**5**

votes

**4**answers

624 views

### The most unexpected and/or the least natural category theory theorem?

Theory of categories is all natural and abstract nonsense. Or is it? What would be the most unexpected and/or the least natural theorem of the theory of category? (It does not really have to be THE).
...

**12**

votes

**9**answers

1k views

### What are your favorite concrete examples of limits or colimits that you would compute during lunch?

(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...

**17**

votes

**30**answers

769 views

### Formalizations of category theory in proof assistants

What are the existing formalizations of category theory in proof assistants?
I'm primarily interested in public-domain code implementing category theory in a proof assistant (Coq, Agda, ...

**12**

votes

**0**answers

414 views

### What are some examples of weak ω-categories?

As is usual, let's say an (n, k)-category is something with
objects, morphisms, 2-morphisms, ..., n-morphisms, such that all
j-morphisms for j > k are invertible, everything meant in the
weak sense. ...

**9**

votes

**4**answers

702 views

### Connections between topos theory and topology

What are some "applications to" / "connections with" topology that one could hope to reasonably cover in a first course on topos theory (for master students)? I have an idea of what parts of the ...

**22**

votes

**2**answers

710 views

### Properties of functors and their adjoints

I am interested in collecting in this question a list of properties a functor $F$ may have and what those properties imply for left and right adjoints, $F^L$ and $F^R$, assuming they exist. There are ...

**6**

votes

**1**answer

625 views

### On the concept of point in category theory

In principle, many different abstractions of the set-theoretic notion of point / element are available in the framework of categories, but they are not equally effective and, what is more interesting ...

**42**

votes

**7**answers

3k 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 ...

**32**

votes

**15**answers

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 ...

**23**

votes

**3**answers

2k views

### In which situations can one see that topological spaces are ill-behaved from the homotopical viewpoint?

In the eighties, Grothendieck devoted a great amount of time to work on the foundations of homotopical algebra.
He wrote in "Esquisse d'un programme": "[D]epuis près d'un an, la plus grande partie ...

**8**

votes

**9**answers

1k views

### “Surprising” categorical equivalences

This is inspired by this question about the equivalence between the category of finite sets and non-negative integers. Now this question was (rightly, I guess) closed, but the fact was surprising to ...

**17**

votes

**3**answers

1k 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 ...

**15**

votes

**12**answers

2k views

### Constructions unique up to non-unique isomorphism

1) Fields have algebraic closures unique up to a non-unique isomorphism.
2) Nice spaces (without base point) have universal covering spaces unique up to a non-unique isomorphism.
3) Modules have ...

**1**

vote

**1**answer

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 ...

**5**

votes

**9**answers

987 views

### What category without initial object do you care about?

Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...

**32**

votes

**5**answers

3k views

### What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of?
I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory.
...