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20
votes
0answers
726 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 ...
14
votes
0answers
226 views

Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?

It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the ...
8
votes
0answers
122 views

A “small” definition of sub-(∞,1)-topoi

Suppose I have an $(\infty,1)$-topos $\mathcal{X}$ and a (small) set of maps $S$ in $\mathcal{X}$, which therefore generates an accessible localization $S^{-1}\mathcal{X}$. Is there any "small" ...
7
votes
0answers
163 views

A “universally non Hypercomplete” $\infty$-topos?

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected ...
7
votes
0answers
468 views

A model category for descent?

Recall that an $(\infty,1)$-category $C$ is said to have descent if for any small diagram $X:I\to M$ with (homotopy) colimit $\overline{X}$, the adjunction between $C/\overline{X}$ and "equifibered" ...
7
votes
0answers
185 views

Lex $\infty$-colimits

In the paper lex colimits, Garner and Lack gave a general characterization of "exactness properties" for categories (and enriched categories). A "lex-weight" is a weight for colimits whose domain ...
7
votes
0answers
364 views

Cohesive ∞-Toposes

Say an $\infty$-topos $\mathbf{H}$ is cohesive if its global section geometric morphism $\Gamma : \mathbf{H} \to \infty \mathrm{Grpd}$ admits a further left adjoint $\Pi$ and a further right adjoint ...
6
votes
0answers
126 views

Is hypercompletion functorial?

Given an infinity topos $\mathcal{E}$, one can pass to its hypercompletion $\widehat{\mathcal{E}}$, which is the full subcategory on those objects local with respect to $\infty$-connected morphisms ...
5
votes
0answers
86 views

Finite covers in complex analytic geometry

Given a complex manifold or complex analytic space, one has the standard notion of open set. There are two different Grothendieck topologies that one can define using this notion, one where covers ...
4
votes
0answers
322 views

Commutation of simplicial homotopy colimits and homotopy products in spaces

Edit: The claim below is wrong, as explained in the comments, because infinite homotopy products of simplicial sets require their components to be fibrantly replaced first, unlike finite homotopy ...
4
votes
0answers
225 views

Is there a local projective model structure on simplicial sheaves? What are its fibrant objects?

Consider a site S (I am mostly interested in hypercomplete sites, e.g., the site of smooth manifolds). The category of simplicial presheaves SPSh(S) on S can be equipped with the local projective ...
3
votes
0answers
171 views

What are the Čech-local equivalences of (simplicial pre)sheaves?

Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield ...
3
votes
0answers
306 views

Which sites in classical/derived algebraic geometry are hypercomplete?

Local questions: 1) Given a commutative ring $A,$ is $Sh_\infty\left(Spec(A)\right)$ hypercomplete? 2) Given a commutative ring $A,$ is $Sh_\infty\left(Et\left(A\right)\right)$ hypercomplete, where ...
2
votes
0answers
154 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 ...
2
votes
0answers
113 views

surjection of localic infinity toposes?

Hello! Is there a simple 'topological' condition to detect whenever a morphism of locales $f : X \rightarrow Y$ induces a surjection of infinity-toposes $f : \mathrm{Sh}_{\infty}(X) \rightarrow ...