On the homotopy theory of stratified spaces. (Sur la théorie homotopique des espaces stratifiés.) (English. French summary) Zbl 07839895
In classical algebraic topology it was shown that there is an equivalence between the homotopy theory of topological spaces and that of simplicial sets. That gave more combinatorial and algebraic tools in the study of homotopy theory of spaces. The paper under review tries to develop a similar theory for stratified spaces.
Let \(P\) be a poset. Write \(\mathbf{Top}^{ex}_{/P} \) for the full subcategory of those \(P\)-stratified topological spaces \(T\) for which Lurie’s exit-path simplicial set \(\text{Sing}_P(T)\) is a quasicategory. The \(\infty\)-categories that arise in this way have a special property that the fibers of the structure morphism to \(P\) are given by the fundamental \(\infty\)-groupoids of the strata. In particular, every morphism in each fiber is invertible. Equivalently, the structure morphism is a conservative functor. Let \(W^P\) denote the class of morphisms that induce \(P\)-weak homotopy equivalences on all strata and topological links. Equivalently, \(W^P\) is the class of morphisms that are sent to equivalences in the Joyal-Kan model structure under \(\text{Sing}_P\). Actually they are morphisms that induce weak homotopy equivalences on strata and on their links. Furthermore the morphisms are Douteau-Henriques equivalences and their image under \(\text{Sing}_P\) is an equivalence when regarded as a morphism in the \(\infty\)-category \(\mathbf{Str}_P\). Then there is an equivalence of the localization \(\mathbf{Top}^{ex}_{/P} [W_P^{-1}]\) and the category \(\mathbf{Str}_P\). Here \(\mathbf{Str}_P\) is the \(\infty\)-category of \(\infty\)-categories over (the nerve) of \(P\) where the structure morphism is conservative. That provides a positive answer to a conjecture of Ayala-Francis-Rozenblyum [D. Ayala et al., J. Eur. Math. Soc. (JEMS) 21, No. 4, 1071–1178 (2019; Zbl 1445.57019)]. In particular the results holds for conically topological stratified spaces like Whitney stratified spaces and stratified spaces in the sense of Goresky-MacPherson,
As a consequence of the main result the author gets that the \(\infty\)-category \(\mathbf{Str}_P\) is equivalent to an \(\omega\)-accessible localization of the underlying \(\infty\)-category of the combinatorial simplicial model category \(\mathbf{Top}_P\) in the Douteau-Henriques model structure.
Let \(P\) be a poset. Write \(\mathbf{Top}^{ex}_{/P} \) for the full subcategory of those \(P\)-stratified topological spaces \(T\) for which Lurie’s exit-path simplicial set \(\text{Sing}_P(T)\) is a quasicategory. The \(\infty\)-categories that arise in this way have a special property that the fibers of the structure morphism to \(P\) are given by the fundamental \(\infty\)-groupoids of the strata. In particular, every morphism in each fiber is invertible. Equivalently, the structure morphism is a conservative functor. Let \(W^P\) denote the class of morphisms that induce \(P\)-weak homotopy equivalences on all strata and topological links. Equivalently, \(W^P\) is the class of morphisms that are sent to equivalences in the Joyal-Kan model structure under \(\text{Sing}_P\). Actually they are morphisms that induce weak homotopy equivalences on strata and on their links. Furthermore the morphisms are Douteau-Henriques equivalences and their image under \(\text{Sing}_P\) is an equivalence when regarded as a morphism in the \(\infty\)-category \(\mathbf{Str}_P\). Then there is an equivalence of the localization \(\mathbf{Top}^{ex}_{/P} [W_P^{-1}]\) and the category \(\mathbf{Str}_P\). Here \(\mathbf{Str}_P\) is the \(\infty\)-category of \(\infty\)-categories over (the nerve) of \(P\) where the structure morphism is conservative. That provides a positive answer to a conjecture of Ayala-Francis-Rozenblyum [D. Ayala et al., J. Eur. Math. Soc. (JEMS) 21, No. 4, 1071–1178 (2019; Zbl 1445.57019)]. In particular the results holds for conically topological stratified spaces like Whitney stratified spaces and stratified spaces in the sense of Goresky-MacPherson,
As a consequence of the main result the author gets that the \(\infty\)-category \(\mathbf{Str}_P\) is equivalent to an \(\omega\)-accessible localization of the underlying \(\infty\)-category of the combinatorial simplicial model category \(\mathbf{Top}_P\) in the Douteau-Henriques model structure.
Reviewer: Stratos Prassidis (Karlovasi)
MSC:
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
| 57N80 | Stratifications in topological manifolds |
| 18N40 | Homotopical algebra, Quillen model categories, derivators |
Citations:
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