nLab pro-homotopy theory



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




Pro-homotopy theory involves the study of model categories and other abstract homotopy theoretic structure on pro-categories of topological spaces or simplicial sets. (The term can also be used for any extension of homotopical structures for a category CC to the corresponding category Pro(C)Pro(C) of pro-objects in CC.)


and is closely related to profinite homotopy theory.

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 TT, Bull. Soc. Math. France, 124, (1996), 347–373,

The initial reference to Quick’s work is :

  • G. Quick, Profinite homotopy theory, PDF

but a correction to an error in the proof of the main result was included in

  • G. Quick, Continuous group actions on profinite spaces, J. Pure Appl. Algebra 215 (2011), 1024-1039.


For one of the earliest model structures, namely the strict model structure on Pro(C)Pro(C), see

  • D.A. Edwards and H. M. Hastings, (1976), Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Maths. 542, Springer-Verlag, pdf

More recent contributions include:

Last revised on September 14, 2021 at 09:21:05. See the history of this page for a list of all contributions to it.