# Contents

## Idea

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 $Pro(C)$.)

## Uses

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

• 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 theorypdf

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.

Revised on July 21, 2014 04:12:22 by Tim Porter (2.26.15.61)