nLab simplicial profinite space

A simplicial profinite space is a simplicial object in the category of profinite spaces (i.e. of Stone spaces).

Simplicial profinite spaces are not equivalent to pro-objects in the category of finite simplicial sets, as was pointed out by Isaksen.

The homotopy theory of simplicial profinite spaces has been explored by Fabien Morel and Gereon Quick.

For Morel’s theory see

• F. Morel, Ensembles profinis simpliciaux et interprétation géométrique du foncteur $T$, Bull. Soc. Math. France, 124, (1996), 347–373,

A reference to Quick’s work is in

• G. Quick, Profinite homotopy theory, pdf

but there was an error in the proof of one of the main results which has been corrected in

• G. Quick, Continuous group actions on profinite spaces, J. Pure Appl. Algebra 215 (2011),

  1024-1039.

(Note there are several incorrect statements, notably with respect to the construction of non-abelian cohomology, in these papers, but the essentials of the proofs would seem to be correct.)