nLab
stratified space

Idea

A notion of stratified space is supposed to be a notion of topological space that is not necessarily a manifold, but which is filtered into “strata” that are.

Examples include polyhedra, algebraic varieties, orbit spaces of many group actions on manifolds and mapping cylinders of maps between manifolds.

Definitions

There are various ways in the literature to make this precise, including:

• stratifolds
• orbifolds
• poset-stratified space
• stratified category

Examples

• the moduli stack of formal groups has a natural stratification by height of formal groups.

Related concepts

• Maybe one can assign a fundamental category to a stratified space. This has been studied for poset-stratified spaces – see there under “exit path $\infty$-category” for more.

• stratification

• piecewise flat spacetime

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

whilst an earlier paper on exit paths is

• David Treumann, Exit paths and constructible stacks, Compositio Math.

  145 (2009), 1504–1532