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infinity-local site

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

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 CC is \infty-local if

Remark

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

Proposition

Theorem

For CC an \infty-local site, the (∞,1)-sheaf (∞,1)-topos Sh (,1)(C)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 (,1)ΓGrpd. 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 (,1)(C)[C op,sSet] proj,loc . 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 Γ\Gamma is given by evaliation on the terminal object. At the level of simplicial presheaves the sSet-enriched right adjoint to Γ\Gamma is given by

S:UsSet(Γ(U),S) \nabla S : U \mapsto sSet(\Gamma(U), S)

as confirmed by the following end/coend calculus computation:

[C op,sSet](X,(S)) = UCsSet(X(U),sSet(Γ(U),S) = UCsSet(X(U)×Γ(U),S) =sSet( UCX(U)×Γ(U),S) =sSet( UCX(U)×Hom C(*,U),S) =sSet(X(*),S) =sSet(Γ(X),S), \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

(Γ):[C op,sSet] projΓsSet Quillen (\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[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 :sSet Quillen[C op,sSet] proj,loc\nabla : sSet_{Quillen} \to [C^{op}, sSet]_{proj,loc} preserves fibrant objects, in that for SS a Kan complex we have that S\nabla S satisfies descent along Cech nerves of covering families.

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

[C op,sSet](U,S)=sSet(Hom C(*,U),S)sSet(Hom C(*,C(U)),S)=[C op,sSet](C(U),S), [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 QuillensSet_{Quillen} is a simplicial model category so that homming the weak equivalence between cofibrant objects into the fibrant object SS indeed yields a weak equivalence (using the factorization lemma).

Examples

and

Revised on January 11, 2011 12:24:54 by Urs Schreiber (89.204.137.68)