Stratified space
Introduction In mathematics, especially in topology, a '''stratified space''' is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat).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 can be defined as a sheaf that is locally constant 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.
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, (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 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*Stratified Morse theory*Harder–Narasimhan stratification
Footnotes
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