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. ...
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 ...
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 ...
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 ...
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 ...
[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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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. ...