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**18**

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### $\infty$-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 ...

**9**

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129 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**

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112 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" ...

**6**

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305 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" ...

**6**

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160 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 ...

**6**

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115 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 ...

**6**

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352 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 ...

**4**

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281 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**

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197 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**

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158 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**

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268 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**

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136 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**

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111 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 ...