nLab cosimplicial simplicial set



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



Paths and cylinders

Homotopy groups

Basic facts




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

ΔsSet \Delta \to sSet

which equivalently is a functor

Δ×Δ opSet. \Delta \times \Delta^{op} \to Set \,.

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


Model structure

There are several standard ways to equip sSet ΔsSet^\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.


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 June 11, 2022 at 16:45:36. See the history of this page for a list of all contributions to it.