Pro-homotopy theory involves the study of model category and other abstract homotopy theoretic structure on pro-categories of spaces or simplicial sets. (The term can also be used for any extension of homotopical structures for a category $C$ to the corresponding $Pro(C)$.)
In étale homotopy theory and motivic homotopy.
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
A reference to Quick’s work is in
but a correction to an error in the proof of the main result was included in
For one of the earliest model structures, namely the strict model structure on $Pro(C)$, see
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 Fausk D. C. Isaksen, Model structures on pro–categories, Homology, Homotopy and Applications, Vol. 9 (2007), 367–398.