cosimplicial simplicial set
Paths and cylinders
A cosimplicial simplicial set is a cosimplicial object in the category of simplicial sets, hence a functor
which equivalently is a functor
There is an eviden model category structure on which models cosimplicial infinity groupoids.
There are several standard ways to equip 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.
The standard reference is
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
Revised on December 5, 2013 00:56:23
by Urs Schreiber