nLab
infinity-connected (infinity,1)-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 being locally and globally \infty-connected means that it satisfies sufficient conditions such that the (∞,1)-category of (∞,1)-sheaves over it is a locally ∞-connected (∞,1)-topos and a ∞-connected (∞,1)-topos.

Definition

Definition

A a site is locally and globally \infty-connected over ∞Grpd if

Properties

Theorem

The (∞,1)-sheaf (∞,1)-topos Sh (,1)(C)Sh_{(\infty,1)}(C) over locally and globally \infty-conneted site CC, regarded as an (∞,1)-site, is a (1-localic) locally ∞-connected (∞,1)-topos and ∞-connected (∞,1)-topos, in that it comes with a triple of adjoint (∞,1)-functors

(ΠΔΓ):Sh (,1)(C)ΓΔΠGrpd (\Pi \dashv \Delta \dashv \Gamma) : Sh_{(\infty,1)}(C) \stackrel{\stackrel{\Pi}{\to}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd

such that Π\Pi preserves the terminal object.

To prove this, we we use the model structure on simplicial presheaves to present Sh (,1)(C)Sh_{(\infty,1)}(C).

Write [C op,sSet] proj[C^{op}, sSet]_{proj} for the projective global model structure and [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc} for its left Bousfield localization at the set of morphisms C({U i})UC(\{U_i\}) \to U out of the Cech nerve for each covering family {U iU}\{U_i \to U\}, and [C op,sSet] proj,loc [C^{op}, sSet]_{proj,loc}^\circ for the Kan complex-enriched category on the fibrant-cofibrant objects. By the discussion at model structure on simplicial presheaves we have

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

and the adjoint (∞,1)-functors on the left are presented by simplicial Quillen adjunctions on the right.

To establish these, we proceed by a sequence of lemmas.

Standard fact

The model categories

are all left proper model categories.

Proof

The first since all objects are cofibrant. The second by general statements about the global model structure on functors, the third because left Bousfield localization preserves left propernes.

Lemma

For {U iU}\{U_i \to U\} a covering family in the \infty-connected site CC, the Cech nerve C({U i})[C op,sSet]C(\{U_i\}) \in [C^{op}, sSet] is a cofibrant resolution of UU both in the projective model structure [C op,sSet] proj[C^{op}, sSet]_{proj} as well as in the Cech local model structure [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc}.

Proof

By assumption on CC we have that C({U i})C(\{U_i\}) is a split hypercover. By Dugger's theorem on cofibrant objects in the projective model structure this implies that C(U)C(U) is cofibrant in the global model structure. By general properties of left Bousfield localization we have that the cofibrations in the local model structure as the same as in the global one. Finally that C({U i})UC(\{U_i\}) \to U is a weak equivalence in the local model structure holds effectively by definition (since we are localizing at these morphisms).

Proposition

On a locally and globally \infty-connected site CC the global section (∞,1)-geometric morphism

(ΔΓ):Sh (,1)(C)ΓΔGrpd (\Delta \dashv \Gamma) : Sh_{(\infty,1)}(C) \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd

is presented by the simplicial Quillen adjunction

(ConstΓ):[C op,sSet] proj,locsSet Quillen, (Const \dashv \Gamma) : [C^{op}, sSet]_{proj,loc} \stackrel{\leftarrow}{\to} sSet_{Quillen} \,,

where Γ\Gamma is the functor that evaluates on the point, ΓX=X(*)\Gamma X = X(*), and ConstConst is the functor that sends a simplicial set SS to the presheaf constant on that value, ConstS:USConst S : U \mapsto S.

Proof

We use (as described there) that adjoint (∞,1)-functors are modeled by simplicial Quillen adjunctions between the simplicial model categories that model the (,1)(\infty,1)-categories in question.

That we have an adjunction (ConstΓ)(Const \dashv \Gamma) follows for instance by observing that since CC has a terminal object we may think of Γ\Gamma as being the functor Γ=lim \Gamma = \lim_\leftarrow that takes the limit.

To see that we have a Quillen adjunction first notice that we have a Quillen adjunction

(ConstΓ):[C op,sSet] projsSet Quillen (Const \dashv \Gamma) : [C^{op}, sSet]_{proj} \stackrel{\leftarrow}{\to} sSet_{Quillen}

on the global model structure, since Γ\Gamma manifestly preserves fibrations and acyclic fibrations there. Since [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc} is left proper and has the same cofibrations as the global model structure, it follows with HTT, corollary A.3.7.2 (see the discussion of sSet-Quillen adjunctions) that for this to descend to a Quillen adjunction on the local model structure it is sufficient that Γ\Gamma preserves fibrant objects. But every fibrant object in the local structure is in particular fibrant in the global structure, hence in particular fibrant over the terminal object of CC.

The left derived functor of Const:sSet Quillen[C op,sSet] projConst : sSet_{Quillen} \to [C^{op},sSet]_{proj} preserves homotopy limits (because (∞,1)-limits in an (∞,1)-category of (∞,1)-presheaves are computed objectwise), and ∞-stackification, the left derived functor of Id:[C op,sSet] proj[C op,sSet] proj,locId : [C^{op}, sSet]_{proj} \to [C^{op}, sSet]_{proj,loc} is a left exact (∞,1)-functor, therefore the left derived functor of Const:sSet Quillen[C op,sSet] proj,locConst : sSet_{Quillen} \to [C^{op}, sSet]_{proj,loc} preserves finite homotopy limits.

This means that our Quillen adjunction does model a (∞,1)-geometric morphism Sh (,1)(C)GrpdSh_{(\infty,1)}(C) \to \infty Grpd. By the discussion at global section the space of these geometric morphisms to ∞Grpd is contractible, hence this is indeed a representative of the terminal geometric morphism as claimed.

Proof of the theorem

By general abstract facts the sSet-functor Const:sSet[C op,sSet]Const : sSet \to [C^{op}, sSet] given on SsSetS \in sSet by Const S:USConst_S : U \mapsto S for all UCU \in C has an sSet-left adjoint

Π:X UX(U)=lim X \Pi : X \mapsto \int^U X(U) = \lim_\to X

naturally in XX and SS, given by the colimit operation. Notice that since sSet is itself a category of presheaves (on the simplex category), these colimits are degreewise colimits in Set. Also notice that the colimit over a representable functor is the point (by a simple Yoneda lemma-style argument).

Regarded as a functor sSet Quillen[C op,sSet] projsSet_{Quillen} \to [C^{op}, sSet]_{proj} the functor ConstConst manifestly preserves fibrations and acyclic fibrations and hence

(ΠConst):[C op,sSet] projConstlim sSet Quillen (\Pi \dashv Const) : [C^{op}, sSet]_{proj} \stackrel{\overset{\lim_\to}{\to}}{\underset{Const}{\leftarrow}} sSet_{Quillen}

is a Quillen adjunction, in particular Π:[C op,sSet] projsSet Quillen\Pi : [C^{op},sSet]_{proj} \to sSet_{Quillen} preserves cofibrations. Since by general properties of left Bousfield localization of model categories the cofibrations of [C op,sSet] proj,loc[C^{op},sSet]_{proj,loc} are the same, also Π:[C op,sSet] proj,locsSet Quillen\Pi : [C^{op}, sSet]_{proj,loc} \to sSet_{Quillen} preserves cofibrations.

Since sSet QuillensSet_{Quillen} is a left proper model category it follows as before with HTT, corollary A.3.7.2 (see the discussion of sSet-Quillen adjunctions) that for

(ΠConst):[C op,sSet] proj,locConstlim sSet Quillen (\Pi \dashv Const) : [C^{op}, sSet]_{proj,loc} \stackrel{\overset{\lim_\to}{\to}}{\underset{Const}{\leftarrow}} sSet_{Quillen}

to be a Quillen adjunction, it suffices to show that ConstConst preserves fibrant objects. That means that constant simplicial presheaves satisfy descent along covering families in the \infty-cohesive site CC: for every covering family {U iU}\{U_i \to U\} in CC and every simplicial set SS it must be true that

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

is a homotopy equivalence of Kan complexes. (Here we use that UU, being a representable, is cofibrant, that C(U)C(U) is cofibrant by the above lemma and that ConstSConst S is fibrant in the projective structure by the assumption that SS is fibrant. So the simplicial hom-complexes in the above equaltion really are the correct derived hom-spaces.)

But that this is the case follows by the condition on the \infty-cohesive site CC by which lim C(U)*\lim_\to C(U) \simeq *: using this it follows that

[C op,sSet](C(U),ConstS)=sSet(lim C(U),S)sSet(*,S)=S. [C^{op}, sSet](C(U), Const S) = sSet(\lim_\to C(U), S) \simeq sSet(*, S) = S \,.

So we have established that also

(ΠConst):[C op,sSet] proj,locConstlim sSet Quillen (\Pi \dashv Const) : [C^{op}, sSet]_{proj,loc} \stackrel{\overset{\lim_\to}{\to}}{\underset{Const}{\leftarrow}} sSet_{Quillen}

is a Quillen adjunction.

It is clear that the left derived functor of Π\Pi preserves the terminal object: since that is representable by assumption on CC, it is cofibrant in [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc}, hence 𝕃lim *=lim *=*\mathbb{L} \lim_\to * = \lim_\to * = *.

Examples

Proposition

The sites

are locally and globally \infty-connected and in fact ∞-cohesive.

This implies that ∞LieGrpd is a cohesive (∞,1)-topos. See there for details.

and

Revised on January 6, 2011 17:58:45 by Urs Schreiber (89.204.137.97)