## A simplicial approach to stratified homotopy theory

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- by Sylvain Douteau PDF
- Trans. Amer. Math. Soc.
**374**(2021), 955-1006 Request permission

## Abstract:

In this article we consider the homotopy theory of stratified spaces from a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. We then define a generalization of the homotopy groups for any fibrant filtered simplicial set $X$ : the filtered homotopy groups $s\pi _n(X)$. They are diagrams of groups built from the homotopy groups of the different pieces of $X$. We then show that the weak equivalences are exactly the morphisms that induce isomorphisms on those filtered homotopy groups.

Then, using filtered versions of the topological realisation of a simplicial set and of the simplicial set of singular simplices, we transfer those results to a category whose objects are topological spaces stratified over $P$. In particular, we get a stratified version of Whitehead’s theorem. Specializing to the case of conically stratified spaces, a wide class of topological stratified spaces, we recover a theorem of Miller saying that to understand the homotopy type of conically stratified spaces, one only has to understand the homotopy type of strata and holinks. We then provide a family of examples of conically stratified spaces and of computations of their filtered homotopy groups.

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## Additional Information

**Sylvain Douteau**- Affiliation: LAMFA CNRS UMR 7352 - Université de Picardie Jules Verne, Amiens, France
- MR Author ID: 1344860
- Received by editor(s): August 4, 2018
- Received by editor(s) in revised form: March 20, 2020
- Published electronically: November 25, 2020
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 955-1006 - MSC (2010): Primary 55U35, 57N80, 18G30, 18G55
- DOI: https://doi.org/10.1090/tran/8264
- MathSciNet review: 4196384