8
votes
2answers
453 views

Modern versions of Verdier's hypercovering theorem?

Let $\mathcal{C}$ be a small category equipped with a terminal object $1$ and a Grothendieck topology. (Assume $\mathcal{C}$ also has pullbacks, if it is more convenient.) The following is a ...
2
votes
0answers
128 views

Cloven Kan fibrations

For each groupoid $G$ there is a monad such that the cloven fibred groupoids over $G$ are algebras for this monad. I am looking for an infinite dimensional counterpart: a monad on simplicial sets ...
6
votes
3answers
373 views

classifying $\infty$-toposes for topological/localic groups?

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ? More precisely, is there an $\infty$-topos $BG$ ...
5
votes
0answers
97 views

Which dense inclusions of sites are ∞-dense?

An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D. Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ...
5
votes
2answers
311 views

Is the site of (smooth) manifolds hypercomplete?

By site of manifolds Man, I mean the category of manifolds (maybe submanifolds to obtain a small category) with continuous maps between them. A Grothendieck topology is given by open covers. Actually, ...
9
votes
1answer
289 views

Boolean non-hypercomplete $(\infty,1)$-toposes

Let's say that an $(\infty,1)$-topos is Boolean if for every object $X$, the lattice $Sub(X)$ of subobjects (i.e. $(-1)$-truncated morphisms into $X$) is a Boolean algebra. I think this is equivalent ...
10
votes
1answer
327 views

Are $\infty$-topoi determined by their localic points ?

Hello ! If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an ...
7
votes
2answers
434 views

Canonical topology for infinity topoi revisited.

A while ago I asked this quetion: Canonical topology for big infinity topoi and this question: How to resolve size issues with the regular epimorphism topology Let me first summarize some of what I ...
17
votes
0answers
619 views

$\infty$-topos and localic $\infty$-groupoids?

It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales). For the record, this is proved by, starting ...