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