nLab model structure on cosimplicial simplicial sets

Idea

For $\Delta$ the simplex category the functor category $sSet^{\Delta}$ is that of cosimplicial objects in simplicial sets: cosimplicial simplicial sets.

There are various standard model category structures on this category. The Reedy model structure is discussed in (BousfieldKan), the injective structure is discussed in (Jardine).

Properties

Remark

The totalization of a cosimplicial simplicial set $X^\bullet$ coincides with the sSet-enriched hom-object

$$Tot(X_\bullet) = sSet^{\Delta}(\Delta, X_\bullet) \,,$$

where $\Delta : [k] \mapsto \Delta[k]$ is the canonical cosimplicial simplicial set given by the simplex-assignment.

Since $\Delta$ is cofibrant in the Reedy model structure it follows that totalization of Reedy-fibrant cosimplicial simplicial sets preserves weak equivalences. The following lists situations in which totalization respects weak equivalences even without this assumption.

Remark

Totalization is closely related to descent objects. If $A$ is a simplicial presheaf and $Y \to X$ is a hypercover, then the descent object is the sSet-enriched hom

$$Desc(Y,X) = sPsh(Y,A) \,.$$

If we decompose

$$Y = \int^{[n] \in \Delta} \Delta[n] \cdot Y_n$$

into its cells by a coend, where now each $Y_n$ is a Set-valued presheaf (see co-Yoneda lemma), then this is

$$\begin{aligned} Desc(Y,A) &= sPSh(\int^{n \in \Delta} \Delta[n] \cdot Y_n, A) \\ & = \int_{n \in \Delta} sPSh(\Delta[n] \cdot Y_n , A) \\ & = \int_{n \in \Delta} sSet(\Delta[n], sPSh(Y_n,A)) \\ &= Tot( sPSh(Y_\bullet, A)) \end{aligned} \,,$$

where the equality signs are isomorphisms of simplicial sets, the outside integral sign denotes the end, and in the integrand we are using that simplicial presheaves are simplicially enriched and tensored over simplicial sets.

So a standard class of examples of cosimplicial simplicial sets to keep in mind are those obtained by evaluating a simplicial presheaf degreewise on the components of a hypercover. Its totalization then is the corresponding descent object.

Proposition

For $G^\bullet \to H^\bullet$ a morphism of cosimplicial groupoids which is degreewise an equivalence, also the induced morphism of totalizations

$$Tot(G^\bullet) \to Tot(H^\bullet)$$

is a weak equivalence (of simplicial sets).

This is (Jardine, corollary 12).

Definition

Let $\Delta_+ \hookrightarrow \Delta$ be the subcategory of the simplex category on the co-face maps. Write $rTot$ for the corresponding totalization, called the restricted totalization.

Proposition

For $G^\bullet \to H^\bullet$ a degreewise weak equivalence of strict 2-groupoids, the resulting morphism of connected components of restricted totalizations

$$rTot(G^\bullet) \to rTot(H^\bullet)$$

is a weak equivalence.

This is (Prezma, theorem 6.1).

Retsricted to $\pi_0$ this statement appeared as (Yekutieli, theorem 2.4). Notice that it is indeed necessary to use the restricted totalization instead of the ordinary totalization here.

References

The Reedy model structure on $sSet^{\Delta}$ is discussed in Chapter X of

• Aldridge Bousfield and Dan Kan, Homotopy limits, completions and localizations Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.

The injective model structure is discussed in

• Rick Jardine, Cosimplicial spaces and cocycles (pdf)

Totalization of cosimplicial strict 2-groupoids is considered in

• Amnon Yekutieli, Combinatorial Descent Data for Gerbes (arXiv:1109.1919)

and

• Matan Prezma, Descent data of cosimplicial 2-groupoids (arXiv:1112.3072)