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

Idea

An (∞,1)-site is locally $\mathrm{\infty }$-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 $(\mathrm{\infty },1)$-presheaf functor has a left adjoint (∞,1)-functor given by taking colimits

$${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)\stackrel{\stackrel{}{\hookrightarrow }}{\underset{L}{\leftarrow }}{\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)\stackrel{\stackrel{\underrightarrow{\lim }}{\to }}{\underset{\mathrm{Const}}{\leftarrow }}\mathrm{\infty }\mathrm{Grpd}.$$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 $(\mathrm{\infty },1)$-presheaf is an $(\mathrm{\infty },1)$-sheaf, this implies that we have also natural equivalences

$$\begin{array}{rl}\mathrm{Sh}(X,L\mathrm{Const}S)& \simeq \mathrm{PSh}(C,\mathrm{Const}S)\\ & \simeq \mathrm{\infty }\mathrm{Grpd}(\underrightarrow{\lim }C,S)\end{array}.$$\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 ${\lim }_{\to }:[C^{\mathrm{op}},\mathrm{sSet}]\to \mathrm{sSet}$ sends this to a weak equivalence

$$\underrightarrow{\lim }Y\stackrel{\simeq }{\to }\underrightarrow{\lim }U=*$$\lim_\to Y \stackrel{\simeq}{\to} \lim_\to U = * \,

Then $C$ is locally $\mathrm{\infty }$-connected.

Proof

We may present ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ by the projective model structure on simplicial presheaves $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$ left Bousfield localized at the Cech nerve projections $C({\coprod }_i{U}_i)\to U$ for each covering family $\{U_i\to U\}$ 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 $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ 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 $[C^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj}}$. Since every split hypercover is cofibrant, it follows that $\mathrm{Const}S$ is an $\mathrm{\infty }$-sheaf precisely if for all $U\in C$ and some split hypercover $Y\to U$ we have that the morphism on derived hom-spaces

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

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

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

and

$$[C^{\mathrm{op}},\mathrm{sSet}](U,\mathrm{Const}S)\simeq S,$$[C^{op}, sSet](U, Const S) \simeq S \,,

so that the condition is that

$$S\to \mathrm{sSet}(\underrightarrow{\lim }Y,S)$$S \to sSet(\lim_\to Y, S)

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

Corollary

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

Proof

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

Related concepts

• locally connected topos / locally ∞-connected (∞,1)-topos

  • connected topos / ∞-connected (∞,1)-topos

  • strongly connected topos / strongly ∞-connected (∞,1)-topos

  • totally connected topos / totally ∞-connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

and

• locally connected site / locally ∞-connected (∞,1)-site

  • connected site / ∞-connected (∞,1)-site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site