nLab pro-homotopy theory

Idea

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 $C$ to the corresponding category $Pro(C)$ of pro-objects in $C$.)

Uses

• In strong shape theory,

• In étale homotopy theory and motivic homotopy.

• In proper homotopy theory

and is closely related to profinite homotopy theory.

Related concepts

• profinite space

• simplicial profinite space

• pro-left adjoint

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,

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.

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

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 (arXiv:math/0511057)

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