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7
votes
0answers
96 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" ...
5
votes
1answer
248 views

Associative Ring Spectra and Derived Completion

So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question: Is ...
4
votes
0answers
213 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
1answer
174 views

Hypercovers of sheaves in classical and quasi-categories

I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...
8
votes
1answer
657 views

Learning higher differential geometry

I have read parts of the motivation on nlab and all the posts on MO I could find on the subject, and by now there are a few questions on my mind. If they trivial for someone who understands the ...
9
votes
0answers
108 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
2answers
470 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
130 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 ...
7
votes
1answer
185 views

Which morphisms of ring spectra are of effective descent for modules?

There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in ...
6
votes
3answers
400 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 $BG$ ...
3
votes
0answers
138 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
1answer
250 views

Relation between hypercompleteness and the property that Cech cohomology calculates sheaf cohomology

Let $C$ be a small site with fibre products. The (injective) Čech model structure on simplicial presheaves $\operatorname{sPre}(C)$ on $C$ presents an $(\infty,1)$-topos and one may ask if this ...
5
votes
0answers
101 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 ...
6
votes
0answers
145 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 ...
13
votes
2answers
594 views

$\infty$-categorical interpretation of type theory

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity ...
4
votes
1answer
289 views

Homotopy left-exactness of a left derived functor

Let $$ F: \mathcal{C} \leftrightarrows \mathcal{D} :G $$ be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors $$ \mathbb{L}F: ...
5
votes
2answers
325 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, ...
4
votes
0answers
170 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 ...
6
votes
1answer
215 views

FIltered colimits of truncated objects in $\infty$-topoi

The bare question: Let $\mathcal{C}$ be an $\infty$-topos, and let $\tau_{\leq 0}\mathcal{C}$ be the subcategory of 0-truncated objects (which is the nerve of an ordinary Grothendieck topos: see HTT ...
3
votes
0answers
244 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 ...
6
votes
0answers
110 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 ...
9
votes
1answer
293 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
336 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 ...
10
votes
1answer
1k views

Forcing in Homotopy Type Theory

I apologize if this question doesn't make any sense. I'll just go ahead and delete it if that's the case. But the question is just the title. Is there a notion of forcing in homotopy type theory? ...
2
votes
0answers
108 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 ...
5
votes
2answers
311 views

Necessary conditions for cofibrancy in global projective model structure on simplicial presheaves

Consider the global projective model category of simplicial presheaves on some category (the category of smooth manifolds is particularly interesting to me). In Section 9.1 of Dugger's paper ...
7
votes
2answers
446 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
627 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 ...
19
votes
2answers
844 views

Homotopy groups of spheres in a $(\infty, 1)$-topos

Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces). You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, ...
5
votes
0answers
340 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 ...