nLab stratified space

Contents

Contents

Idea

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.

Variants

There are many different definitions of stratifications on spaces:

Basic features

Stratification structures

Most of the above variants share the same basic structure, captured by the following definitions.

Terminology

Given a decomposition ff of a topological space XX into connected subspaces (denoted, in the following, by lower-case letters such as ss), the exit path preorder Exit(f)\mathsf{Exit}(f) of ff is the preorder of subspaces ss in ff with a generating arrow srs \to r whenever the closure of rr intersects ss non-trivially.

Definition

A stratification ff on XX is a decomposition of XX such that Exit(f)\mathsf{Exit}(f) is a poset.

The subspaces in a stratification are also called strata. The opposite poset Exit(f) op\mathsf{Exit}(f)^{\mathrm{op}} is also called the entrance path poset and denoted by Entr(f)\mathsf{Entr}(f).

Remark

Given a stratification ff on XX, there is a map XExit(f)X \to \mathrm{Exit}(f) mapping points xsx \in s to sExit(f)s \in \mathrm{Exit}(f), which is sometimes called the characteristic map of ff, and (abusing notation) denoted by f:XExit(f)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.

Example

(Poset stratifications). Let XX be a space, PP a poset, and f:XPf : X \to P a continuous map (in other words, ff is a PP-stratification of X X ). This determines a stratification c(f)\mathrm{c}(f) of XX whose strata are the connected components of the preimages f 1(x)f^{-1}(x), xPx \in P. The map ff factors uniquely through the characteristic map c(f):XExit(c(f))\mathrm{c}(f) : X \to \mathsf{Exit}(\mathrm{c}(f)) by a conservative poset map Exit(c(f))P\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.

Example

(Filtered spaces). Any filtered space X 0X 1...X nX_0 \subset X_1 \subset ... \subset X_n in which X iX_i is a closed subspace of X i+1X_{i+1} defines a continuous map X[n]=(01...n)X \to [n] = (0 \to 1 \to ... \to n) mapping points in X i+1X iX_{i+1} \setminus X_{i} to ii, 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.

Example

(Trivial stratification). Every topological space UU is trivially stratified with strata being the connected components of UU.

Remark

(Continuity of characteristic map). Let (X,f)(X,f) be a stratified space. One says that the stratification ff is “locally finite” if each stratum ss of ff has an open neighborhood in XX which only contains finitely many strata. (If (X,f)(X,f) satisfies the frontier condition, see next remark, then, equivalently, (X,f)(X,f) is locally finite iff each point xXx \in X has an open neighborhood intersecting only finitely many strata.) If ff is locally finite, then the characteristic map f:XExit(f)f : X \to \mathsf{Exit}(f) is a continuous map.

Remark

(Openness of characteristic map). Let (X,f)(X,f) be a stratified space. One says the stratification ff satisfies the “frontier condition” (or, as an adjective, that it is “frontier-constructible”) if, for any two strata s,rs, r, the closure r¯\overline r intersects ss non-trivially, then sr¯s \subset \overline r. The stratification ff is frontier-constructible iff the characteristic map f:XExit(f)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.

The category of stratifications

Definition

A stratified map F:(X,f)(Y,g)F : (X,f) \to (Y,g) of stratified spaces is a continuous map F:XYF : X \to Y which factors through the characteristic maps ff and gg by a necessarily unique map, denoted by Exit(F):Exit(f)Exit(g)\mathsf{Exit}(F) : \mathsf{Exit}(f) \to \mathsf{Exit}(g).

Stratified spaces and their maps form the category Strat\mathbf{Strat} of stratification. The construction of exit path posets yields a functor Exit:StratPos\mathsf{Exit} : \mathbf{Strat} \to \mathbf{Pos} (dually, using () op:PosPos(-)^{\mathrm{op}} : \mathbf{Pos} \to \mathbf{Pos} one obtains the entrance path poset functor Entr:StratPos\mathsf{Entr} : \mathbf{Strat} \to \mathbf{Pos}). The functor has a right inverse, as follows.

Definition

Every poset PP has a classifying stratification P\parallel P \parallel (also called the stratified realization of PP), whose underlying space is the classifying space |P||P| of PP (i.e. the realization of the nerve of PP), and whose characteristic map is the map |P|P|P| \to P that maps |P x||P <x||P^{\leq x}| \setminus |P^{\lt x}| to xx (here, the full subposets P x={yx}P^{\leq x} = \{y \leq x\} and P <x={y<x}P^{\lt x} = \{y \lt x\} of PP are the “lower” resp. “strict lower closures” of an element xx in PP). Moreover, given a poset map F:PQF : P \to Q, the realization of its nerve yields a stratified map F:PQ\parallel F\parallel : \parallel P\parallel \to \parallel Q \parallel. We obtain a functor :PosStrat\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.

Terminology

(Types of stratified maps). Let F:(X,f)(Y,g)F : (X,f) \to (Y,g) be a stratified map. The stratified map FF is called:

  • a substratification if F:XYF : X \to Y is a subspace and Exit(F)\mathsf{Exit}(F) is conservative; if, moreover, X=g 1Exit(F)f(X)X = g^{-1} \circ \mathsf{Exit}(F) \circ f (X) then one says the substratification is constructible
  • a coarsening if F:XYF : X \to Y is a homeomorphism (to emphasize the opposite process, one also calls FF a refinement);
  • a stratified homeomorphism (or stratified iso) if F:XYF : X \to Y is a homeomorphism of spaces and Exit(F)\mathsf{Exit}(F) is an isomorphism of posets.

Fundamental categories

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-conceptpresentation
sets \simeq (0,0)(0,0)-categories∞-sets \simeq spacessets with w.e.
posets \simeq (0,1)(0,1)-categories∞-posets \simeq stratified spacesposets with w.e.
categories == (1,1)(1,1)-categories∞-categoriescategories 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<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 11-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.

For conical stratificiations

Definition

Given stratifications (X,f)(X,f) and (Y,g)(Y,g) their product is the stratification of X×YX \times Y with characteristic map f×gf \times g.

Definition

Given a stratification (X,f)(X,f), the stratified (open) cone (cone(X),cone(f))(\mathrm{cone}(X), \mathrm{cone}(f)) stratifies the topological open cone cone(X)=X×[0,1)/X×{0}\mathrm{cone}(X) = X \times [0,1) / X \times \{0\} by the product (X,f)×(0,1)(X,f) \times (0,1) away from the cone point {0}\{0\} (here, the open interval (0,1)(0,1) is trivially stratified), and by setting the cone point {0}\{0\} to be its own stratum.

Definition

A conical stratification (X,f)(X,f) is a stratification in which each point xXx \in X has a neighborhood (i.e. an open substratification) that is a stratified product U×(cone(Z),cone(l))U \times (\mathrm{cone}(Z),\mathrm{cone}(l)) for some “link” stratification (Z,l)(Z,l) and such that xU×{0}x \in U \times \{0\}.

Every conical stratification is frontier-constructible.

Definition

Given a conical stratification (X,f)(X,f), then Lurie constructs exit path \infty-category xit(f)\mathcal{E}\mathrm{xit}(f) as a quasicategory: the kk-simplices of the quasicategory xit(f)\mathcal{E}\mathrm{xit}(f) are precisely stratified maps [k](X,f)\parallel [k]\parallel \to (X,f), where [k]=(01...k)[k] = (0 \to 1 \to ... \to k).

The construction translates from “stratified spaces” to “\infty-posets” in the above table: the conservative functor xit(f)Exit(f)\mathcal{E}\mathrm{xit}(f) \to \mathsf{Exit}(f) takes objects [0]X\parallel [0]\parallel \to X to the stratum sExit(f)s \in \mathsf{Exit}(f) that their image lies in.

For regular stratifications

Definition

A stratification (X,f)(X,f) is regular if it admits a refinement P(X,f)\parallel P \parallel \to (X,f) by the stratified realization of some poset PP.

Definition

Given a regular stratification (X,f)(X,f) and a refinement F:P(X,f)F : \parallel P\parallel \to (X,f), one can construct the presented exit path \infty-category 𝒫ℰxit(f)\mathcal{PE}\mathrm{xit}(f) as a poset with weak equivalences with underlying poset PP and weak equivalences Exit(F) 1(id)\mathsf{Exit}(F)^{-1}(\mathrm{id}).

(Showing that this construction is, in an appropriate sense, independent of the choice of PP requires a bit more work…)

Remark

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”.

References

A notion of purely topologically stratified sets was introduced in

  • Frank Quinn, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), 441–499. MR 89g:57050

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

  • L. Siebenmann, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 47 (1971), 123–165. MR 47:7752

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

  • Bruce Hughes, Geometric topology of stratified spaces (pdf)

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

based on the notion of stratifications developed in

  • David Ayala, John Francis, and Hiro Lee Tanaka. Local structures on stratified spaces. Advances in Mathematics 307 (2017): 903-1028.

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:

The perspective of stratifications as spatial decompositions that determine exit path posets and characteristic maps can be found e.g. in:

  • Christoph Dorn and Christopher Douglas, Framed combinatorial topology, 2021, in particular Appendix B (pdf)

Last revised on September 24, 2022 at 13:56:22. See the history of this page for a list of all contributions to it.