It seems that the theory of constructible sheaves (in particular anything that goes into proving that they form an abelian category) requires some technical statements about existence of certain stratifications and yet at the same time the notion of a "stratified space" seems to have several inequivalent definitions in the literature.

  1. Is there (as of today) an agreement among specialists in the field for what is the "correct" definition of a "stratified space"?

  2. What would be a good reference to read about stratified spaces, constructible sheaves and their six functors? (in particular a complete construction of the abelian and derived categories of constructible sheaves). (Perhaps with a little bit of stratified Morse theory too).

My end goal is to have a concrete understanding of perverse sheaves on algebraic varieties and their stratifications. I don't want to get bogged down in the technicalities of stratified spaces but I would like to know how stratifications look and how to construct them in this context. However if there's a "natural theory" (and in particular a natural category) of stratified spaces I'd love to read about it.


2 Answers 2


Consult the book "Sheaves on manifolds" by Kashiwara and Schapira. It's a hard nut to crack, but it is the most efficient presentation I've seen.

In Chapter 8, they work with stratifications whose strata satisfy the $\mu$-regularity condition. This is equivalent to the Verdier regularity condition which in turn is a bit stronger than the Whitney condition b.

One can show that any subanalytic set admits stratifications satisfying the $\mu$-regularity conditions. In particular, this condition implies that such stratified spaces admit (many) stratified Morse functions.

  • $\begingroup$ I'll look into it. This might be a different question but is there a characteristic free way to establish the theory of constructible and perverse sheaves? $\endgroup$ Jan 3, 2017 at 10:21
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    $\begingroup$ See the same reference. $\endgroup$ Jan 3, 2017 at 13:03
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    $\begingroup$ Maybe I wasn't clear but the book doesn't seem to mention anything about algebraic varieties over fields of positive characteristic. $\endgroup$ Jan 4, 2017 at 15:42
  • $\begingroup$ When you referred to characteristic, I thought you where referring to the fields of coefficients of the various sheaves. jmilne.org/math/CourseNotes/LEC.pdf $\endgroup$ Jan 4, 2017 at 16:34

A bit more reader friendly than Kashiwara-Schapira is Borel's Intersection Cohomology. It doesn't treat perverse sheaves, but it has a good overview of stratified spaces in the first few chapter and develops constructible sheaves and the six functors in Chapter V. You might also try Dimca's Sheaves in Topology, which does talk about perverse sheaves. Neither of these present the most general possible development, but they're pretty accessible.

I'll also plug the discussion of stratified spaces in my book-in-progress, though I take a more topological approach that's not particularly oriented toward sheaves. Here's a link: http://faculty.tcu.edu/gfriedman/IHbook.pdf


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