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:

  • I. Barnea and T. M. Schlank, 2011, A Projective Model Structure on

    Pro Simplicial Sheaves, and the Relative Etale Homotopy Type_, arXiv:1109.5477

  • I. Barnea and T. M. Schlank, 2013, Functorial Factorizations in Pro

    Categories_, arXiv:1305.4607.

  • D. C. Isaksen, A model structure on the category of pro-simplicial sets,

    Trans. Amer. Math. Soc., 353, (2001), 2805–2841.

  • D. C. Isaksen, Calculating limits and colimits in pro-categories, Fundamenta Mathematicae, 175, (2002), 175 – 194.

  • D. C. Isaksen, 2004, Strict model structures for pro-categories , in Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001) , volume 215 of Progr. Math., 179 – 198, Birkhauser, Basel.

  • D. C. Isaksen, Completions of pro-spaces , Math. Z., 250, (2005), 113

    – 143.

  • Halvard FauskD. C. Isaksen, Model structures on pro–categories, Homology, Homotopy and Applications, Vol. 9 (2007), 367–398.

  • Ilan Barnea, Yonatan Harpaz, Geoffroy Horel, Pro-categories in homotopy theory (arXiv:1507.01564)

Last revised on April 22, 2020 at 22:39:09. See the history of this page for a list of all contributions to it.