is there a sensible notion of "abelian $\infty$-category"?

@BrunoStonek I'd argue that prestable ∞-categories are the correct analogue. See also SAG.C.5.4 for a notion of "abelian n-category" for every n, giving abelian categories for n=1 and prestable ∞-categories for n=∞

thanks for the pointer, I'll take a look

The whole appendix C of SAG is relevant

Has anyone ever thought about the space of good covers on a manifold? In particular what is the homotopy type of the space of good covers of the open n-ball? A good cover is a finite union of open sets diffeomorphic to open balls s.t. all possible intersections are again a finite disjoint union of open sets diffeomorphic to open n-balls.

How do good covers form a space?

Also, I thought the definition was: the finite intersections of elements of the cover are either balls or empty.

@CharlesRezk I imagine there are many ways to describe the topology. I could order all the open sets in the cover than give a metric induced from the L^2 metric on characteristic functions of the open sets, then mod out by the permutation action on the indices of the ordering. I imagine there are way more simpler ways to phrase this topology and I think any other reasonable thing would give something which is at least homotopic. I may be wrong though.

As for the definition of good, I don't really care too much whether or not the definition is as you stated or as I stated. i would be interested in the answer regardless.

And I also agree that the definition I gave is non standard.

btw, i'm really interested in the chat history but i don't have access to the dropbox yet, how do I get in with all the cool kids? Is there a secret password?

Hello all ! First time here (=

Does anyone know about the current state of results regarding a version of Ravenel's localization conjecture in the motivic setting ?

I'm not a cool kid either, so I dunno