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 $Pro(C)$.)

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.

References

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

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.