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Stream: deprecated: topos theory

Topic: classifying spaces of categories

Morgan Rogers (he/him) (Apr 14 2020 at 11:37):

Following on from but tangential to this thread, if I correctly understand the results in Section 5.1.4 of Lurie's Higher Topos Theory, the operation of idempotent-completion induces a homotopy equivalence on classifying spaces. That fact is very significant to me, because an immediate consequence is that the homotopy type of a small category is an invariant of its presheaf topos. I can naively recover the homotopy type of a category by taking the classifying space of the subcategory indecomposable projective objects, but one is led to wonder:

1. Is there a less rigid construction which might apply, if not in all Grothendieck toposes, at least in a larger class of them? For example, is there some explicit way to relate the category of sheaves on the classifying space, or perhaps more combinatorially the slice topos of simplicial sets over the nerve of $\mathcal{C}$, to $[\mathcal{C}^{\mathrm{op}},\mathrm{Set}]$?
2. What can be said about the classifying space of $\mathcal{C}$ given only a property of the topos $[\mathcal{C}^{\mathrm{op}},\mathrm{Set}]$, and vice versa?

I anticipate that some people will argue that this is exactly the kind of question that $\infty$-toposes are built to handle. If so, please gently guide me towards the answers that they might hold to this question.

Pierre Cagne (Apr 20 2020 at 08:36):

Not sure if this is what you're are after, but there is always a map of topos $\mathrm{Sh}(BC) \to \hat C$ (where $BC$ is the topological realization of the nerve of $C$) which can be shown to be a Artin-Mazur equivalence. This is actually a way of showing the following characterization of weak equivalences in $\mathbf{Cat}$: a functor $u:\hat A \to \hat B$ is a weak equivalence if and only if the induced topos-morphism $\hat A \to \hat B$ is a Artin-Mazur equivalence.

Morgan Rogers (he/him) (Apr 20 2020 at 08:59):

Where map of toposes is geometric morphism? And "Artin-Mazur equivalence" means some version of homotopy equivalence? And $u$ in the last sentence should be $A \to B$ rather than $\hat{A} \to \hat{B}$? Interesting! How is the morphism $\mathrm{Sh}(BC) \to \hat{C}$ defined?

Mike Shulman (Apr 20 2020 at 21:56):

A reference that I like for this sort of thing is Moerdijk's Classifying spaces and classifying topoi.

Morgan Rogers (he/him) (Apr 21 2020 at 09:32):

Thanks, I hadn't heard of this before, I'll have a read!