nLab infinity-local site

Idea

A site is $\infty$-local if it satisfies sufficient conditions for the (∞,1)-sheaf (∞,1)-topos over it to be a local (∞,1)-topos.

Definition

Definition

A site $C$ is $\infty$-local if

• it has a terminal object $*$;

• the limit-functor $\lim_\leftarrow : [C^{op}, sSet] \to$ sSet sends Cech nerve projections $C(U) \to X$ over covering families $\{U_i \to X\}$ to weak homotopy equivalences:

  $$Hom_C(*, C(U)) \stackrel{\simeq}{\to} Hom_C(*,X) \,.$$

Remark

If $C$ is also a strongly ∞-connected site then it is an ∞-cohesive site.

Proposition

Theorem

For $C$ an $\infty$-local site, the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over it is a local (∞,1)-topos, in that the global section (∞,1)-geometric morphism has a further right adjoint (∞,1)-functor

$$Sh_{(\infty,1)} \stackrel{\overset{\nabla}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{\nabla}{\leftarrow}}} \infty Grpd \,.$$

Proof

We may present the (∞,1)-sheaf (∞,1)-topos by the local model structure on simplicial presheaves

$$Sh_{(\infty,1)}(C) \simeq [C^{op},sSet]_{proj,loc}^\circ \,.$$

For the notation see the details of the analagous proof at ∞-connected site. As discussed there, the functor $\Gamma$ is given by evaliation on the terminal object. At the level of simplicial presheaves the sSet-enriched right adjoint to $\Gamma$ is given by

$$\nabla S : U \mapsto sSet(\Gamma(U), S)$$

as confirmed by the following end/coend calculus computation:

$$\begin{aligned} [C^{op}, sSet](X, \nabla(S)) & = \int_{U \in C} sSet(X(U), sSet(\Gamma(U), S) \\ & = \int_{U \in C} sSet(X(U) \times \Gamma(U), S) \\ & = sSet( \int^{U \in C} X(U) \times \Gamma(U), \;\; S ) \\ & = sSet( \int^{U \in C } X(U) \times Hom_C(*, U), \;\; S) \\ & = sSet(X(*), S) \\ & = sSet(\Gamma(X), S) \end{aligned} \,,$$

where in the second but last step we used the co-Yoneda lemma.

It is clear that

$$(\Gamma \dashv \nabla) : [C^{op}, sSet]_{proj} \stackrel{\overset{\Gamma}{\to}}{\underset{\nabla}{\leftarrow}} sSet_{Quillen}$$

is a Quillen adjunction, since $\nabla$ manifestly preserves fibrations and acyclic fibrations. Since $[C^{op}, sSet]_{proj,loc}$ is a left proper model category to see that this descends to a Quillen adjunction on the local model structure on simplicial presheaves it is sufficient to check that $\nabla : sSet_{Quillen} \to [C^{op}, sSet]_{proj,loc}$ preserves fibrant objects, in that for $S$ a Kan complex we have that $\nabla S$ satisfies descent along Cech nerves of covering families.

This follows from the second defining condition on the $\infty$-local site $C$, that $Hom_C(*,C(U)) \simeq Hom_C(*,U)$. Using this we have for fibrant $S \in sSet_{Quillen}$ the descent weak equivalence

$$[C^{op}, sSet](U, \nabla S) = sSet(Hom_C(*,U), S) \simeq sSet(Hom_C(*,C(U)), S) = [C^{op}, sSet](C(U), \nabla S) \,,$$

where we use in the middle step that $sSet_{Quillen}$ is a simplicial model category so that homming the weak equivalence between cofibrant objects into the fibrant object $S$ indeed yields a weak equivalence (using the factorization lemma).

Examples

• CartSp, ThCartSp

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 site

  • connected site / ∞-connected site

  • strongly connected site / strongly ∞-connected site

  • totally connected site / totally ∞-connected site

• local site / ∞-local site

• cohesive site, ∞-cohesive site