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



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

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos



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



Call an (∞,1)-site CC locally contractible if every constant (∞,1)-presheaf on it is an (∞,1)-sheaf in the (∞,1)-topos over CC.

More explicitly, this means that every covering sieve RR on an object UCU\in C, regarded as a subcategory of C/UC/U, is weakly contractible, i.e. its nerve N(R)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 XGpdX\in \infty Gpd is that the map Const(X)(U)=Xlim RConst(X)Const(X)(U) = X \to \lim_R Const(X) is an equivalence, but lim RConst(X)=Map(N(R),X)\lim_R Const(X) = Map(N(R),X), and this is equivalent to XX for all XX if and only if N(R)N(R) is contractible as an \infty-groupoid.



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

Sh (,1)(C)LPSh (,1)(C)Constlim Grpd. 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 (,1)(\infty,1)-presheaf is an (,1)(\infty,1)-sheaf, this implies that we have also natural equivalences

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



Let CC be an 1-site such that every object UU has a split hypercover YUY \to U such that contracting all representables to points yields a weak equivalence. Equivalently, if the colimit functor lim :[C op,sSet]sSet\lim_\to : [C^{op}, sSet] \to sSet sends this to a weak equivalence

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

Then CC is locally \infty-connected.


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

It is immediate that we have a Quillen adjunction (limconst)(\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 YUY \to U is a cofibrant resolution in [C op,sSet] proj,loc[C^{op}, sSet]_{proj, loc} of UU.

For SsSetS \in sSet a Kan complex let ConstS:C opsSetConst S : C^{op} \to sSet the corresponding constant simplicial presheaf. This is fibrant in [C op,sSet] proj[C^{op}, sSet]_{proj}. Since every split hypercover is cofibrant, it follows that ConstSConst S is an \infty-sheaf precisely if for all UCU \in C and some split hypercover YUY \to U we have that the morphism on derived hom-spaces

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


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

so that the condition is that

SsSet(lim Y,S) S \to sSet(\lim_\to Y, S)

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


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


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


Last revised on February 21, 2020 at 10:18:50. See the history of this page for a list of all contributions to it.