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

Idea

An (∞,1)-site is locally $\infty$-connected if it has properties that ensure that 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 (∞,1)-topos over $C$.

More explicitly, this means that every covering sieve $R$ on an object $U\in C$, regarded as a subcategory of $C/U$, is weakly contractible, i.e. its nerve $N(R)$ (which is just itself, if it is incarnated as a quasicategory) is contractible in the Kan-Quillen model structure on simplicial sets. For the sheaf condition for a constant presheaf on $X\in \infty Gpd$ is that the map $Const(X)(U) = X \to \lim_R Const(X)$ is an equivalence, but $\lim_R Const(X) = Map(N(R),X)$, and this is equivalent to $X$ for all $X$ if and only if $N(R)$ is contractible as an $\infty$-groupoid.

Properties

Proposition

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

$$Sh_{(\infty,1)}(C) \underoverset{\hookrightarrow}{\overset{L}{\longleftarrow}}{\bot} PSh_{(\infty,1)}(C) \stackrel{ \overset{\lim_\to}{\longrightarrow} } { \underset{Const}{\leftarrow} } \infty Grpd \,.$$

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

$$\begin{aligned} Sh(X, L Const S) &\simeq PSh(X, Const S) \\ & \simeq \infty Grpd(\lim_\to X , S) \end{aligned} \,.$$

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 $\lim_\to : [C^{op}, sSet] \to sSet$ sends this to a weak equivalence

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

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

Proof

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

It is immediate that we have a Quillen adjunction $(\underset{\rightarrow}{\lim} \dashv const)$ for the global model structure on simplicial presheaves on both sides. Now by the recognition theorem for simplicial Quillen adjunctions for this to descend to a Quillen adjunction on the local model structure it is sufficient that the left adjoint preserves the cofibrations of the local model structure and (already) that the right adjoint preserves the fibration objects. Since left Bousfield localization of model categories does not change the cofibrations, the first of these is immediate.

This means that to establish the claim it is now sufficient to show that constant simplicial presheaves already satisfy descent for a locally $\infty$-connected site. This is what we do now.

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

For $S \in sSet$ a Kan complex let $Const S : C^{op} \to sSet$ the corresponding constant simplicial presheaf. This is fibrant in $[C^{op}, sSet]_{proj}$. Since every split hypercover is cofibrant, it follows that $Const S$ 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

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

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

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

and

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

so that the condition is that

$$S \to sSet(\lim_\to Y, S)$$

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

Corollary

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

Proof

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