---
tag: 3F84AMVJNV
kind: concept
name: Whitney stratification
msc: [32S60, 58A35, 57N80]
sources:
  - url: https://en.wikipedia.org/wiki/Whitney_conditions
---

## Summary

A Whitney stratification of a singular space is a partition into smooth manifolds, the strata, see the [[definition|F19BQTN832]].
The strata fit together in a controlled way along their boundaries.
The control is given by Whitney's conditions (a) and (b) on limits of tangent planes and secant lines.
Hassler Whitney proved that every complex [[analytic set|3RWWZYPAFV]] has such a stratification, and the result was later extended to semialgebraic and subanalytic sets.
By the Thom–Mather isotopy theorem, a Whitney stratified space is topologically locally trivial along each stratum.
This makes these stratifications the standard setting for [[perverse sheaves|5WHXRZDZFK]] and intersection homology.

## Explanations

### L1 human

A singular space, such as a cone or a surface with a crease, can be cut into smooth pieces of different dimensions: the smooth part, the crease, the tip.
Whitney's conditions ask that the large pieces approach the small ones nicely, without spiralling or twisting in a way that changes the local picture.
Then the space looks the same near every point of a given piece.
For the cone $x^2 + y^2 = z^2$, the pieces are the two halves of the cone without its tip, and the tip.