homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 $C$.)
In étale homotopy theory and motivic homotopy.
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
The initial reference to Quick’s work is :
but a correction to an error in the proof of the main result was included in
For one of the earliest model structures, namely the strict model structure on $Pro(C)$, see
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)
Last revised on September 14, 2021 at 09:21:05. See the history of this page for a list of all contributions to it.