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The term stratified space usually refers to a topological space equipped with the structure of a stratification, which decomposes the space into subspaces called strata. Often strata are “nice” in some sense (for instance, one may require strata to be manifolds, while the stratified space itself need not be a manifold), and the way that strata are linked together is controlled by additional conditions. More generally, stratified spaces may refer to any notion of spaces equipped with some type of stratification structure.
Examples are ubiquitous: they include polyhedra, cell complexes, algebraic varieties, orbit spaces of many group actions on manifolds, mapping cylinders of maps between manifolds, and moduli stacks of formal groups stratified by height of formal groups.
There are many different definitions of stratifications on spaces:
Most of the above variants share the same basic structure, captured by the following definitions.
Given a decomposition $f$ of a topological space $X$ into connected subspaces (denoted, in the following, by lower-case letters such as $s$), the exit path preorder $\mathsf{Exit}(f)$ of $f$ is the preorder of subspaces $s$ in $f$ with a generating arrow $s \to r$ whenever the closure of $r$ intersects $s$ non-trivially.
A stratification $f$ on $X$ is a decomposition of $X$ such that $\mathsf{Exit}(f)$ is a poset.
The subspaces in a stratification are also called strata. The opposite poset $\mathsf{Exit}(f)^{\mathrm{op}}$ is also called the entrance path poset and denoted by $\mathsf{Entr}(f)$.
Given a stratification $f$ on $X$, there is a map $X \to \mathrm{Exit}(f)$ mapping points $x \in s$ to $s \in \mathrm{Exit}(f)$, which is sometimes called the characteristic map of $f$, and (abusing notation) denoted by $f : X \to \mathrm{Exit}(f)$.
The characteristic map need not be continuous in general (unless the stratification is locally finite, see the next remark); it is, however, continuous on the preimage of any finite full subposet of the exit path poset. It is often convenient to construct stratifications by constructing their characteristic map.
(Poset stratifications). Let $X$ be a space, $P$ a poset, and $f : X \to P$ a continuous map (in other words, $f$ is a $P$-stratification of $X$). This determines a stratification $\mathrm{c}(f)$ of $X$ whose strata are the connected components of the preimages $f^{-1}(x)$, $x \in P$. The map $f$ factors uniquely through the characteristic map $\mathrm{c}(f) : X \to \mathsf{Exit}(\mathrm{c}(f))$ by a conservative poset map $\mathsf{Exit}(\mathrm{c}(f)) \to P$. (Such (characteristic,conservative)-factorizations are essentially unique.)
The example shows that any poset-stratification determines a unique stratification. However, many poset-stratifications may determine the same stratification in this way.
(Filtered spaces). Any filtered space $X_0 \subset X_1 \subset ... \subset X_n$ in which $X_i$ is a closed subspace of $X_{i+1}$ defines a continuous map $X \to [n] = (0 \to 1 \to ... \to n)$ mapping points in $X_{i+1} \setminus X_{i}$ to $i$, and thus a stratification by the previous example. As a concrete instance of this example, the filtration by skeleta of any cell complex defines the “stratification by cells” in this way.
(Trivial stratification). Every topological space $U$ is trivially stratified with strata being the connected components of $U$.
(Continuity of characteristic map). Let $(X,f)$ be a stratified space. One says that the stratification $f$ is “locally finite” if each stratum $s$ of $f$ has an open neighborhood in $X$ which only contains finitely many strata. (If $(X,f)$ satisfies the frontier condition, see next remark, then, equivalently, $(X,f)$ is locally finite iff each point $x \in X$ has an open neighborhood intersecting only finitely many strata.) If $f$ is locally finite, then the characteristic map $f : X \to \mathsf{Exit}(f)$ is a continuous map.
(Openness of characteristic map). Let $(X,f)$ be a stratified space. One says the stratification $f$ satisfies the “frontier condition” (or, as an adjective, that it is “frontier-constructible”) if, for any two strata $s, r$, the closure $\overline r$ intersects $s$ non-trivially, then $s \subset \overline r$. The stratification $f$ is frontier-constructible iff the characteristic map $f : X \to \mathsf{Exit}(f)$ is an open map.
It is generally very reasonable to assume stratifications to be locally finite and frontier-constructible.
A stratified map $F : (X,f) \to (Y,g)$ of stratified spaces is a continuous map $F : X \to Y$ which factors through the characteristic maps $f$ and $g$ by a necessarily unique map, denoted by $\mathsf{Exit}(F) : \mathsf{Exit}(f) \to \mathsf{Exit}(g)$.
Stratified spaces and their maps form the category $\mathbf{Strat}$ of stratification. The construction of exit path posets yields a functor $\mathsf{Exit} : \mathbf{Strat} \to \mathbf{Pos}$ (dually, using $(-)^{\mathrm{op}} : \mathbf{Pos} \to \mathbf{Pos}$ one obtains the entrance path poset functor $\mathsf{Entr} : \mathbf{Strat} \to \mathbf{Pos}$). The functor has a right inverse, as follows.
Every poset $P$ has a classifying stratification $\parallel P \parallel$ (also called the stratified realization of $P$), whose underlying space is the classifying space $|P|$ of $P$ (i.e. the realization of the nerve of $P$), and whose characteristic map is the map $|P| \to P$ that maps $|P^{\leq x}| \setminus |P^{\lt x}|$ to $x$ (here, the full subposets $P^{\leq x} = \{y \leq x\}$ and $P^{\lt x} = \{y \lt x\}$ of $P$ are the “lower” resp. “strict lower closures” of an element $x$ in $P$). Moreover, given a poset map $F : P \to Q$, the realization of its nerve yields a stratified map $\parallel F\parallel : \parallel P\parallel \to \parallel Q \parallel$. We obtain a functor $\parallel -\parallel : \mathbf{Pos} \to \mathbf{Strat}$.
Every classifying stratification is frontier-constructible.
It makes sense to further terminologically distinguish maps of stratifications as follows.
(Types of stratified maps). Let $F : (X,f) \to (Y,g)$ be a stratified map. The stratified map $F$ is called:
Just as spaces have fundamental $\infty$-groupoids, stratified spaces also have “fundamental categories”. However, the role of sets for spaces is now played by posets: the following table illustrates the analogy. (The table is further explained below.)
base concept | $\infty$-concept | presentation |
---|---|---|
sets $\simeq$ $(0,0)$-categories | ∞-sets $\simeq$ spaces | sets with w.e. |
posets $\simeq$ $(0,1)$-categories | ∞-posets $\simeq$ stratified spaces | posets with w.e. |
categories $=$ $(1,1)$-categories | ∞-categories | categories with w.e. |
In the table, an “$\infty$-X” is intuitively to be understood as an (∞,∞)-category which admits a conservative functor to an X, where X can e.g. stand for “set”, “poset”, or “category”. Yet more generally, X can be an (n,r)-category for $n,r \lt \infty$. A “set with weak equivalences” means a poset with weak equivalences in which each arrow is a weak equivalence. The left column is related to the middle column by an “$\infty$-zation functor” (which simply includes $1$-structures into $\infty$-structures), and the middle and right columns are related by an “∞-localization functor” (which should be a weak equivalence in some sense).
In order to make the above precise, one must work with sufficiently convenient stratifications. We describe two simple ways of constructing/presenting fundamental $\infty$-posets below.
Given stratifications $(X,f)$ and $(Y,g)$ their product is the stratification of $X \times Y$ with characteristic map $f \times g$.
Given a stratification $(X,f)$, the stratified (open) cone $(\mathrm{cone}(X), \mathrm{cone}(f))$ stratifies the topological open cone $\mathrm{cone}(X) = X \times [0,1) / X \times \{0\}$ by the product $(X,f) \times (0,1)$ away from the cone point $\{0\}$ (here, the open interval $(0,1)$ is trivially stratified), and by setting the cone point $\{0\}$ to be its own stratum.
A conical stratification $(X,f)$ is a stratification in which each point $x \in X$ has a neighborhood (i.e. an open substratification) that is a stratified product $U \times (\mathrm{cone}(Z),\mathrm{cone}(l))$ for some “link” stratification $(Z,l)$ and such that $x \in U \times \{0\}$.
Every conical stratification is frontier-constructible.
Given a conical stratification $(X,f)$, then Lurie constructs exit path $\infty$-category $\mathcal{E}\mathrm{xit}(f)$ as a quasicategory: the $k$-simplices of the quasicategory $\mathcal{E}\mathrm{xit}(f)$ are precisely stratified maps $\parallel [k]\parallel \to (X,f)$, where $[k] = (0 \to 1 \to ... \to k)$.
The construction translates from “stratified spaces” to “$\infty$-posets” in the above table: the conservative functor $\mathcal{E}\mathrm{xit}(f) \to \mathsf{Exit}(f)$ takes objects $\parallel [0]\parallel \to X$ to the stratum $s \in \mathsf{Exit}(f)$ that their image lies in.
A stratification $(X,f)$ is regular if it admits a refinement $\parallel P \parallel \to (X,f)$ by the stratified realization of some poset $P$.
Given a regular stratification $(X,f)$ and a refinement $F : \parallel P\parallel \to (X,f)$, one can construct the presented exit path $\infty$-category $\mathcal{PE}\mathrm{xit}(f)$ as a poset with weak equivalences with underlying poset $P$ and weak equivalences $\mathsf{Exit}(F)^{-1}(\mathrm{id})$.
(Showing that this construction is, in an appropriate sense, independent of the choice of $P$ requires a bit more work…)
A special case of the definition asks for refinements by regular cell complexes (or simplicial complexes) in place of classifying stratifications of posets, in which case one speaks of cellulable (resp. triangulable) stratifications.
The construction translates between “stratified spaces” and “posets with weak equivalences” in the above table. Since spaces are trivially stratified spaces, this specializes to a translation between “spaces” and “sets with weak equivalences”.
A notion of purely topologically stratified sets was introduced in
Notions of smoothly stratified spaces were considered by
H. Whitney, Local properties of analytic varieties, Differentiable and combinatorial topology (S. Cairns, ed.), Princeton Univ. Press, Princeton, 1965, pp. 205–244. MR 32:5924
R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–282. MR 39:970
J. Mather, Notes on topological stability, Harvard Univ., Cambridge, (1970)
Locally conelike stratified spaces have been considered in
These are all special cases of (Quinn’s definition).
Discussion of differential operators on stratified spaces is in
R. B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 217–234, Math. Soc.
Pierre Albin, Eric Leichtnam, Rafe Mazzeo, Paolo Piazza, The signature package on Witt spaces, I. Index classes (arXiv:0906.1568v2)
See also
Discussion of the fundamental category of a (Whitney‑)stratified space is in
Jonathan Woolf, Transversal homotopy theory (arXiv:0910.3322)
M. Banagl, Topological invariants of stratified spaces, Springer Monographs in Math. 2000.
A homotopy hypothesis for stratified spaces is discussed in
David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis, arXiv
Peter J. Haine, On the homotopy theory of stratified spaces (arXiv:1811.01119)
The former is based on the notion of stratifications developed in
An earlier paper on exit paths is
Poset-stratified spaces and the conicality condition, as well as the construction of the fundamental $\infty$-poset of a conical stratification as a quasicategory, first appeared in:
Last revised on March 14, 2023 at 12:29:53. See the history of this page for a list of all contributions to it.