# nLab locally infinity-connected (infinity,1)-site

### Context

#### $\left(\infty ,1\right)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

An (∞,1)-site is locally $\infty$-connected if it has properties that ensure that the hypercompletion of the (∞,1)-category of (∞,1)-sheaves over it is a locally ∞-connected (∞,1)-topos

## Definition

###### Definition

Call an (∞,1)-site $C$ locally contractible if every constant (∞,1)-presheaf on it is an (∞,1)-sheaf in the hypercomplete (∞,1)-topos over $C$.

## Properties

###### Proposition

By the general notion of (∞,1)-colimit the constant $\left(\infty ,1\right)$-presheaf functor has a left adjoint (∞,1)-functor given by taking colimits

${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)\stackrel{\stackrel{}{↪}}{\underset{L}{←}}{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(C\right)\stackrel{\stackrel{\underset{\to }{\mathrm{lim}}}{\to }}{\underset{\mathrm{Const}}{←}}\infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$Sh_{(\infty,1)}(C) \stackrel{ \overset{}{\hookrightarrow} } { \underset{L}{\leftarrow} } PSh_{(\infty,1)}(C) \stackrel{ \overset{\lim_\to}{\to} } { \underset{Const}{\leftarrow} } \infty Grpd \,.

Since the (∞,1)-category of (∞,1)-sheaves sits by a full and faithful (∞,1)-functor inside presheaves and by assumption that every constant $\left(\infty ,1\right)$-presheaf is an $\left(\infty ,1\right)$-sheaf, this implies that we have also natural equivalences

$\begin{array}{rl}\mathrm{Sh}\left(X,L\mathrm{Const}S\right)& \simeq \mathrm{PSh}\left(C,\mathrm{Const}S\right)\\ & \simeq \infty \mathrm{Grpd}\left(\underset{\to }{\mathrm{lim}}C,S\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\begin{aligned} Sh(X, L Const S) &\simeq PSh(C, Const S) \\ & \simeq \infty Grpd(\lim_\to C , S) \end{aligned} \,.

## Examples

###### Proposition

Let $C$ be an 1-site such that every object $U$ has a split hypercover $Y\to U$ such that contracting all representables to points yields a weak equivalence. Equivalently, if the colimit functor ${\mathrm{lim}}_{\to }:\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\to \mathrm{sSet}$ sends this to a weak equivalence

$\underset{\to }{\mathrm{lim}}Y\stackrel{\simeq }{\to }\underset{\to }{\mathrm{lim}}U=*\phantom{\rule{thinmathspace}{0ex}}$\lim_\to Y \stackrel{\simeq}{\to} \lim_\to U = * \,

Then $C$ is locally $\infty$-connected.

###### Proof

We may present ${\mathrm{Sh}}_{\left(\infty ,1\right)}\left(C\right)$ by the projective model structure on simplicial presheaves $\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}$ left Bousfield localized at the Cech nerve projections $C\left({\coprod }_{i}{U}_{i}\right)\to U$ for each covering family $\left\{{U}_{i}\to U\right\}$ in $C$.

By the discussion of cofibrant replacement at model structure on simplicial presheaves we have that a split hypercover $Y\to U$ is a cofibrant resolution in $\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj},\mathrm{loc}}$ of $U$.

For $S\in \mathrm{sSet}$ a Kan complex let $\mathrm{Const}S:{C}^{\mathrm{op}}\to \mathrm{sSet}$ the corresponding constant simplicial presheaf. This is fibrant in $\left[{C}^{\mathrm{op}},\mathrm{sSet}{\right]}_{\mathrm{proj}}$. Since every split hypercover is cofibrant, it follows that $\mathrm{Const}S$ is an $\infty$-sheaf precisely if for all $U\in C$ and some split hypercover $Y\to U$ we have that the morphism on derived hom-spaces

$\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\left(U,\mathrm{Const}S\right)\to \left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\left(Y,\mathrm{Const}S\right)$[C^{op}, sSet](U, Const S) \to [C^{op}, sSet](Y, Const S)

is a weak equivalence (of Kan complexes, necessatily). But we have

$\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\left(Y,\mathrm{Const}S\right)\simeq \mathrm{sSet}\left(\underset{\to }{\mathrm{lim}}Y,S\right)$[C^{op}, sSet](Y, Const S) \simeq sSet(\lim_\to Y, S)

and

$\left[{C}^{\mathrm{op}},\mathrm{sSet}\right]\left(U,\mathrm{Const}S\right)\simeq S\phantom{\rule{thinmathspace}{0ex}},$[C^{op}, sSet](U, Const S) \simeq S \,,

so that the condition is that

$S\to \mathrm{sSet}\left(\underset{\to }{\mathrm{lim}}Y,S\right)$S \to sSet(\lim_\to Y, S)

is a weak equivalence. This is the case for all $S$ precisely if ${\mathrm{lim}}_{\to }S$ is contractible, which is precisely our assumption on $Y$.

###### Corollary

Let $X$ be a locally contractible topological space. Then ${\stackrel{^}{\mathrm{Sh}}}_{\left(\infty ,1\right)}\left(C\right)$ is a locally ∞-connected (∞,1)-topos.

###### Proof

The category of open subsets $\mathrm{Op}\left(X\right)$ is not in general a locally $\infty$-connected site according to the above definition. But there is another site of definition for ${\stackrel{^}{\mathrm{Sh}}}_{\left(\infty ,1\right)}\left(X\right)$ which is: the full subcategory $\mathrm{cOp}\left(X\right)↪\mathrm{Op}\left(X\right)$ on the contractible open subsets.

and

Revised on January 15, 2011 10:40:39 by Urs Schreiber (89.204.153.96)