nLab cosimplicial simplicial set

Definition

A cosimplicial simplicial set is a cosimplicial object in the category of simplicial sets, hence a functor

$$\Delta \to sSet$$

which equivalently is a functor

$$\Delta \times \Delta^{op} \to Set \,.$$

There is an evident model category structure on $sSet^{\Delta}$ which models cosimplicial infinity groupoids.

Properties

Model structure

There are several standard ways to equip $sSet^\Delta$ with the structure of a model category. See model structure on cosimplicial simplicial sets for more.

Homotopy and homology

The homotopy groups of the totalization of a cosimplicial homotopy type are computed by a Bousfield-Kan spectral sequence. The homology groups by an Eilenberg-Moore spectral sequence.

References

The standard reference is

• Paul Goerss, Rick Jardine, Simplicial homotopy theory

Chapter X of

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

The homotopy spectral sequence for cosimplicial spaces is in chapter VIII.

• Rick Jardine, Cosimplicial spaces and cocycles (pdf)

• Thomas Goodwillie, A remark on the homology of cosimplicial spaces , Journal of Pure and Applied Algebra Volume 127, Issue 2, 15 May 1998, Pages 167-175