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
Stratification may refer to:
Mathematics
* Stratification (mathematics), any consistent assignment of numbers to predicate symbols
* Data stratification in statistics
Earth sciences
* Stable and unstable stratification
* Stratification, or st ...
, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat
Flattening stratification
/ref>).
A basic example is a subset of a smooth manifold that admits a Whitney stratification In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965.
A stratification of a topological space is a finite filtration by closed sub ...
. 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 origi ...
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
"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the sequence of logically linked ideas in his impor ...
'' considers (or proposes) a stratified space with what he calls the tame topology.
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 E ...
, (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 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 are su ...
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
*Perverse sheaf The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space ''X'', which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was intro ...
* Stratified Morse theory
* Harder–Narasimhan stratification
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