nLab
model structure on cosimplicial simplicial sets

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

For Δ\Delta the simplex category the functor category sSet ΔsSet^{\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 X^\bullet coincides with the sSet-enriched hom-object

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

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

Since Δ\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 AA is a simplicial presheaf and YXY \to X is a hypercover, then the descent object is the sSet-enriched hom

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

If we decompose

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

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

Desc(Y,A) =sPSh( nΔΔ[n]Y n,A) = nΔsPSh(Δ[n]Y n,A) = nΔsSet(Δ[n],sPSh(Y n,A)) =Tot(sPSh(Y ,A)), \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 H G^\bullet \to H^\bullet a morphism of cosimplicial groupoids which is degreewise an equivalence, also the induced morphism of totalizations

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

is a weak equivalence (of simplicial sets).

This is (Jardine, corollary 12).

Definition

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

Proposition

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

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

is a weak equivalence.

This is (Prezma, theorem 6.1).

Retsricted to π 0\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 ΔsSet^{\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

Totalization of cosimplicial strict 2-groupoids is considered in

and

Revised on September 5, 2013 00:55:32 by Anonymous Coward (146.96.35.189)