nLab poset-stratified space

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 $X\to P$, where $X$ is an arbitrary topological space and $P$ is a $T_0$ 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 $X\to Y$ and a poset map $P\to Q$ making a square commute.

Examples

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

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

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

Conically smooth atlases and the $\infty$-category $Strat$

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 $Strat$.

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

Exit path $\infty$-categories

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

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

using the above embedding of $\Delta$ into $Strat$. 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 $Exit : \mathcal{S}trat \hookrightarrow Cat_\infty$, and that certain sheaves on $\mathcal{S}$trat, known as striation sheaves, are equivalent to $(\infty, 1)$-categories.

Related concepts

• stratified category

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

• Peter J. Haine, On the homotopy theory of stratified spaces (arXiv:1811.01119)

whilst an earlier paper on exit paths is

• David Treumann, Exit paths and constructible stacks, Compositio Math.145 (2009), 1504–1532