This is a reduced version of a preliminary (draft) Profinite Algebraic Homotopy (8 chapters).
The idea grew out of two different problems back in the early 1980s. My student, Fahmi Korkes, wrote a beautiful thesis on it and we published two papers. Based on his thesis I started this ‘monograph’.
Two of the motivating problems, at that time, were:
Investigation showed that the theory of profinite group cohomology had certain lacunae in it at the time and these, to some extent, could be filled by adopting ‘crossed’ technology.
Much of the theory goes through painlessly, but once in a while you have to work hard!
Comments of referees on an early draft of these notes suggested including a treatment (or at least a mention) of the work of Isaksen on model category structures on pro-categories and also the theory of simplicial profinite spaces in work by Morel and Quick. As Quick’s published papers on this have some errors, and miss out some important details, a fresh look at the Morel - Quick theory is being developed, using ideas in some of Isaksen’s papers.
The new methods will look at fibrantly generated model category structures, including the class fibrantly generated model category structures of Chorny. These are clearly closely related to pro-categories and the duals of accessible categories.
I hope also to be able to provide an overview of the work of Mandell on $p$-adic homotopy theory and then the links with infinity category theory as in work of Lurie, (DAG XIII).
For one of the earliest model structures on pro-categories, namely the strict model structure on $Pro(C)$, see
More recent contributions to pro-homotopy theory 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.
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