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My understanding is that definition 2. does not usually give a good notion of composition. Why? In categories of sets and quasicoherent sheaves you can compose $\alpha \subset X\times Y\ $ with $ …

I'm learning about representable functors from Vistoli notes thanks to Charles Siegel's answer. I see that any category $\mathcal C$ can be embedded into $\text{Hom}\\,(\mathcal C^{op}, \mathcal Set) …

Every question asked can be divided into two parts: what is known, and what is asked. I think your question's "what is known" part is by no means universally agreed. It's not frustrating to hear peop …

I learned basic category theory from some good book (I forgot which one) and more advanced stuff from Gelfand-Manin. I never encountered the problem of I don't know any way of learning category theory …

I think for mathematicians, especially the algebraic geometers, category theory has a somewhat different meaning that in your area. For us, it's primarily an important and quite straightforward way t …

I have sometimes hard time reading papers that are written in the language of schemes being replaced by the functors they represent (I have especially homotopy scheme theory in mind). I think the to …

Extensions exist for non-abelian groups too.

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 K …

(1) Yes, I think that's one of the ways to define schemes. Look for representable functors and you'll get lots of literature. It was a crazy idea about 50 years go, part of establishment nowadays. …

Some time ago I was also puzzled by this same question, and only now, after seeing yours, I start to think maybe I understand the idea. These are my own thoughts though, so you're encouraged to re-che …