nLab
poset-stratified space

Spaces stratified by a poset

Spaces stratified by a poset

Idea

A poset-stratified space is a particular way to define a stratified space that is convenient for some purposes.

Definition

A poset-stratified space is a continuous map XPX\to P, where XX is an arbitrary topological space and PP is a T 0T_0 Alexandroff space, i.e. a poset with the Alexandroff topology.

The category of stratified spaces is a full subcategory of the arrow category of Top. Note that Alexandroff topologies embed Pos fully-faithfully in Top, so a map of stratified spaces consists of a continuous map XYX\to Y and a poset map PQP\to Q making a square commute.

Examples

  • The cone on a stratified space XPX\to P is the quotient X×[0,1)/X×{0}X\times [0,1) / X\times \{0\}, stratified by the poset P P^\lhd that adjoins a new bottom element to PP.

  • The standard stratification of the nn-simplex Δ n\Delta^n is obtained by regarding it as the (n+1)(n+1)-fold cone of \emptyset. It is stratified by the poset [n]={0,,n}[n] = \{0,\dots,n\}. Note that every stratified map Δ nΔ m\Delta^n \to \Delta^m has an underlying poset map [n][m][n]\to [m]; it is a theorem the resulting map Strat(Δ n,Δ m)Pos([n],[m])Strat(\Delta^n,\Delta^m) \to Pos([n],[m]) is a homotopy equivalence. Thus, we have an embedding of the simplex category into StratStrat.

  • Any manifold MM can be equipped with the trivial stratification over the terminal poset. As a special case, the extended simplices Δ e n={t n+1:t i=1}\Delta^n_e = \{ \vec{t} \in \mathbb{R}^{n+1} : \sum t_i = 1\} assemble into a cosimplicial object Δ e cStrat\Delta^\bullet_e \in cStrat.

Conically smooth atlases and the \infty-category StratStrat

There is a notion of a conically smooth atlas on a stratified space. (The definition is quite involved, inducting on a number of parameters simultaneously; see (Ayala–Francis–Tanaka, section 3).) The category of conically smooth stratified spaces (with conically smooth maps among them) is often again simply called StratStrat.

One can endow this new category StratStrat with an enrichment in Kan complexes via the extended simplices, by defining map Strat(X,Y) =hom Strat(X×Δ e ,Y)\mathrm{map}_{Strat}(X,Y)_\bullet = \mathrm{hom}_{Strat}(X \times \Delta^\bullet_e,Y). This presents the \infty-categorical localization 𝒮trat\mathcal{S}trat of StratStrat at the stratified homotopy equivalences (see (Ayala–Francis–Rozenblyum, Theorem 2.4.5)).

Exit path \infty-categories

For a stratified space XX, its exit path \infty-category is a simplicial space (in the \infty-categorical sense) defined by

Exit(X) p=hom 𝒮trat(Δ p,X)Exit(X)_p = \hom_{\mathcal{S}trat}(\Delta^p,X)

using the above embedding of Δ\Delta into StratStrat. It is proven in Ayala-Francis-Rozenblyum that this is indeed a complete Segal space. The main result of that paper is the stratified homotopy hypothesis, which is the assertion that this construction defines a fully-faithful embedding Exit:𝒮tratCat Exit : \mathcal{S}trat \hookrightarrow Cat_\infty, and that certain sheaves on 𝒮\mathcal{S}trat, known as striation sheaves, are equivalent to (,1)(\infty, 1)-categories.

References

  • Jacob Lurie, Higher Algebra, Appendix A.5

  • David Ayala, John Francis, Hiro Lee Tanaka, Local structures on stratified spaces, arXiv

  • David Ayala, John Francis, Nick Rozenblyum, Factorization homology from higher categories, arXiv

  • David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis, arXiv

whilst an earlier paper on exit paths is

Last revised on September 18, 2018 at 09:31:05. See the history of this page for a list of all contributions to it.