Tim Porter profinite algebraic homotopy

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:

• Combinatorial group theory for profinite groups: The theory that had been developed by Brown and Huebschmann on Identities among Relations and which used both topological methods and algebraic combinatorial group theoretic ones, gave neat descriptions of identities among relations. This had applications to the construction of smaller (crossed) resolutions. It could not be used as such for profinite groups since the topological constructions involved construction of spaces from presentations of a (discrete) group and there seemed no obvious way to generalise this to profinite presentations of profinite groups. Could the construction of free crossed modules etc. that made up the other half of the Brown-Huebschmann approach be adapted to that profinite context?

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.

• Profinite Algebraic Homotopy Theory? The Grothendieck approach to the fundamental groups of schemes produced profinite groups, (cf. SGA1). His Pursuing Stacks manuscript and his earlier letter to Breen suggested that there was a higher homotopy theory lurking around. This was also suggested by the Artin-Mazur lecture notes, Étale homotopy, (SLN 100). $n$-Stacks were related to crossed modules and their higher analogues, so was there a profinite version of crossed modules, and all the related crossed homological algebra?

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).

References

For one of the earliest model structures on pro-categories, 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

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

• F. Morel, Ensembles profinis simpliciaux et interprétation géométrique du foncteur $T$, Bull. Soc. Math. France, 124, (1996), 347–373,

A 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.