5
$\begingroup$

Is there a reasonably well-behaved $2$-category of abelian categories, and if so, what are its objects (perhaps one needs to restrict one's attention to finite abelian categories ?), $1$-morphisms (perhaps exact functors ?), and $2$-morphisms ? In particular, would such a $2$-category be closed under (co)directed $2$-(co)limits ?

Furthermore, would we expect such a category to admit sub-$2$-categories of AB3, AB4, etc. categories ? Why or why not ?

Improve this question
$\endgroup$
5
  • 2
    $\begingroup$ Yes, there's such a 2-category. Objects are finitely presentable abelian categories, and morphism categories are formed by those continuous functors that are sheaves w.r.t. suitable Grothendieck topology. Essentially this is Gabriel-Popescu theorem. Main selling point for this choice of morphisms is that morphism categories are abelian themselves. But then I can ask OP: what exactly do you need this for? (sheer curiosity is always a good reason, but meaning of "reasonably well-behaved" can vary wildly from case to case) $\endgroup$
    – Denis T
    Mar 6, 2022 at 1:14
  • 1
    $\begingroup$ ...to expand a little: morphism categories being abelian means that you can take "derived category of everything" in this picture (not really, but almost), and it gives right answer for Hochschild (over a field) or Mac Lane (in absolute setting) homology purposes. $\endgroup$
    – Denis T
    Mar 6, 2022 at 1:35
  • 2
    $\begingroup$ It depends on what you want, in many cases Rex (finitely cocomplete categories with right exact functors) or Lex (finitely complete categories with left exact functors) will be better behaved. For example see this paper. $\endgroup$
    – Noah Snyder
    Mar 6, 2022 at 2:04
  • 1
    $\begingroup$ take a look here arxiv.org/pdf/1202.0426.pdf $\endgroup$
    – display llvll
    Mar 6, 2022 at 6:18
  • $\begingroup$ @DenisT. Would you mind explaining why the existence of such a 2-category of abelian categories is essentially the Gabriel-Popescu Theorem ? and what was the Grothendieck topology that you mentioned ? What are some references on this topic that you'd recommend ? Also, to answer your question, my interest in this topic is more-or-less due only to curiosity. I'm in the possibility of studying abelian categories in a manner similar to how we study abelian groups (i.e. through homomorphisms, kernels, cokernels, etc.). $\endgroup$
    – Dat Minh Ha
    Mar 6, 2022 at 22:02

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.