nLab
cosimplicial simplicial set
Contents
Context
Homotopy theory
homotopy theory , (∞,1)-category theory , homotopy type theory

flavors: stable , equivariant , rational , p-adic , proper , geometric , cohesive , directed …

models: topological , simplicial , localic , …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Contents
Definition
A cosimplicial simplicial set is a cosimplicial object in the category of simplicial set s, 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 groupoid s.

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

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

Last revised on October 4, 2019 at 08:33:47.
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