In mathematics, especially in topology, a stratified space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
that admits or is equipped with a
stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat
Flattening stratification
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A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space.
On a stratified space, a constructible sheaf In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space ''X'', such that ''X'' is the union of a finite number of locally closed subsets on each of which the sheaf is a locally constant sheaf. It has its ori ...
can be defined as a sheaf that is locally constant
In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. ...
on each stratum.
Among the several ideals, Grothendieck's '' Esquisse d’un programme'' considers (or proposes) a stratified space with what he calls the tame topology
In mathematics, a tame topology is a hypothetical topology proposed by Alexander Grothendieck in his research program '' Esquisse d’un programme'' under the French name ''topologie modérée'' (moderate topology). It is a topology in which the ...
.
A stratified space in the sense of Mather
Mather gives the following definition of a stratified space. A ''prestratification'' on a topological space ''X'' is a partition of ''X'' into subsets (called strata) such that (a) each stratum is locally closed In topology, a branch of mathematics, a subset E of a topological space X is said to be locally closed if any of the following equivalent conditions are satisfied:
* E is the intersection of an open set and a closed set in X.
* For each point x\in ...
, (b) it is locally finite and (c) (axiom of frontier) if two strata ''A'', ''B'' are such that the closure of ''A'' intersects ''B'', then ''B'' lies in the closure of ''A''. A ''stratification'' on ''X'' is a rule that assigns to a point ''x'' in ''X'' a set germ
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functi ...
at ''x'' of a closed subset of ''X'' that satisfies the following axiom: for each point ''x'' in ''X'', there exists a neighborhood ''U'' of ''x'' and a prestratification of ''U'' such that for each ''y'' in ''U'', is the set germ at ''y'' of the stratum of the prestratification on ''U'' containing ''y''.
A stratified space is then a topological space equipped with a stratification.
Pseudomanifold
In the MacPherson's stratified pseudomanifolds; the strata are the differences ''Xi+i-Xi'' between sets in the filtration. There is also a local conical condition; there must be an almost smooth atlas where locally each little open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...
looks like the product of two factors ''Rnx c(L)''; a euclidean factor and the topological cone of a space ''L''. Classically, here is the point where the definitions turns to be obscure, since ''L'' is asked to be a stratified pseudomanifold. The logical problem is avoided by an inductive trick which makes different the objects ''L'' and ''X''.
The changes of charts or cocycles have no conditions in the MacPherson's original context. Pflaum asks them to be smooth, while in the Thom-Mather context they must preserve the above decomposition, they have to be smooth in the Euclidean factor and preserve the conical radium.
See also
*Equisingularity In algebraic geometry, an equisingularity is, roughly, a family of singularities that are not non-equivalent and is an important notion in singularity theory. There is no universal definition of equisingularity but Zariki's equisingularity is the ...
* Perverse sheaf
*Stratified Morse theory In mathematics, stratified Morse theory is an analogue to Morse theory for general stratified spaces, originally developed by Mark Goresky and Robert MacPherson. The main point of the theory is to consider functions f : M \to \mathbb R and con ...
*Harder–Narasimhan stratification In algebraic geometry and complex geometry, the Harder–Narasimhan stratification is any of a stratification of the moduli stack of principal ''G''-bundles by locally closed substacks in terms of "loci of instabilities". In the original form due ...
Footnotes
{{reflist
References
*Appendix 1 of R. MacPherson, Intersection homology and perverse sheaves, 1990 notes
*J. Mather, Stratifications and Mappings, Dynamical Systems, Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26–August 14, 1971, 1973, pages 195–232.
*Markus J. Pflaum, Analytic and Geometric Study of Stratified Spaces: Contributions to Analytic and Geometric Aspects (Lecture Notes in Mathematics, 1768) ; Publisher, Springer;
Further reading
*https://ncatlab.org/nlab/show/stratified+space
*https://mathoverflow.net/questions/258562/correct-definition-of-stratified-spaces-and-reference-for-constructible-sheave
*Chapter 2 of Greg Friedman
Singular intersection homology
* https://ncatlab.org/nlab/show/poset-stratified+space
Stratifications
Topology