derived smooth geometry
There are various ways in the literature to make this precise.
Among them are
A notion of purely topologically stratified sets was introduced in
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
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)
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.