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stable (infinity,1)-topos

A stable $(\mathrm{\infty }\mathrm{,1})$-topos is to an (∞,1)-topos as a stable (∞,1)-category is to an (∞,1)-category.

Recall that a Grothendieck-Rezk-Lurie (∞,1)-topos is an (∞,1)-category of (∞,1)-sheaves, i.e. of sheaves with values in the (∞,1)-category of ∞-groupoids.

A stable Grothendieck–Rezk–Lurie (∞,1)-topos should therefore be an (∞,1)-category of sheaves with values in spectra.

But maybe a full formalization of this idea has not appeared yet (?)

Mike: While the general study of sheaves of spectra is certainly an important topic, I’m not sure I agree that they should be called “stable $(\mathrm{\infty }\mathrm{,1})$-toposes.” They’re not very topos-like; they’re much more like, say, Ab.

Urs: okay, right. We should remove this entry then, or do something to it. But let’s see why I created it, that may indicate a question I have which would be nice to answer in some form:

over at cohomology I said something like: see, in every $(\mathrm{\infty }\mathrm{,1})$-category $H$ which behaves roughly like Top does, we are entitled to think of the hom-spaces $H(X,A)$ as computing cohomology of the source object $X$ with coefficients in the target object $A$;

if the objects of $H$ moreover are stable, then this reproduces abelian cohomology.

so I wanted to put this together and say: if it looks stable and homs in it can be thought of as computing cohomology, then it should be called stable $(\mathrm{\infty }\mathrm{,1})$-topos. Now, I see, that may be a wrong idea. But don’t we want to say something like that? Or should we think of the hom spaces of every possible $(\mathrm{\infty }\mathrm{,1})$-category as some kind of cohomology? That seems wrong, too. (?)

Toby: You could call this an abelian $(\mathrm{\infty }\mathrm{,1})$-category. A topos is like the category of sets ($0$-groupoids), while an abelian category is like the category of abelian groups (stably monoidal $0$-groupoids). So we just change $0$ to $(\mathrm{\infty }\mathrm{,1})$, equivalently $1$ to $(\mathrm{\infty }\mathrm{,1})$, in both of those.

Mike: Well, while an abelian category is like the category of abelian groups, there are plenty of abelian categories other than categories of abelian sheaves, and it sounds like Urs is particularly interested in $(\mathrm{\infty }\mathrm{,1})$-categories of sheaves of spectra.

Also, I feel like a broken record saying this, but the category of spectra is not the category of stably monoidal $\mathrm{\infty }$-groupoids; that would be the category of connective spectra.

Toby: I didn't think that this would make a difference to the elementary properties of the category, but maybe that's wrong. I certainly see your point about restricting to categories of sheaves, however; I expected that that would go under the ‘Grothendieck’ or ‘Grothendieck–Rezk–Lurie’ condition. But do we even have yet a notion of ‘Grothendieck’ abelian category that reproduces the concept of a category of abelian sheaves (other than simply ‘category of abelian shaves’)?

Mike: Including the non-connective spectra makes a big difference to the elementary properties: otherwise the “suspension” functor would not be an auto-equivalence. I don’t think I’ve ever seen a set of Giraud-like axioms for categories of abelian sheaves, but it seems plausible that they might exist. I’m nothing like an expert on abelian categories.

Toby: All right. Well, it might be possible to get something out of ‘abelian’ yet, but presumably not ‘abelian $(\mathrm{\infty }\mathrm{,1})$-category’ exactly.

Mike: Maybe the best notion of “abelian” in the $(\mathrm{\infty }\mathrm{,1})$-world is just a stable (∞,1)-category.