nLab
cosimplicial simplicial set

Definition

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

$$\Delta \to \mathrm{sSet}$$\Delta \to sSet

which equivalently is a functor

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

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

Properties

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

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