# Cohomology of constructible sheaves via exit paths

Let $$X$$ be a stratified space, with stratification $$S$$ (we will ignore technicalities).

The category of exit paths $$Ex(X,S)$$ is a directed refinement of the path groupoid of $$X$$ accounting for the stratification. More precicely the objects of $$Ex(X,S)$$ are points of $$X$$, and morphisms between points are stratified homotopy classes of paths in $$X$$ which do monotone with respect to strata. By stratified homotopy classes I mean that the homotopies themselves are monotone in strata in a suitable way.

It is observed by MacPherson (unpublished) that the category of $$S$$ constructible sheaves of sets on $$X$$ is equivalent to the category from functors from $$Ex(X,S)$$ to the category of sets. I assume that this extends to constructible sheaves in vector spaces, but I don't know a reference.

This fact, together with the equivalence between local systems and $$\pi_{1}(X)$$-representations, tells us that a constructible sheaf of vector spaces is the same as a local system for each stratum together with morphisms of local systems encoding how the strata fit together in $$X$$.

Does this interpretation have any use in the study of cohomology with coefficients in constructible sheaves? More precisely, given a constructible sheaf $$F$$ on $$X$$, can you recover the cohomology groups with coefficients in $$F$$ from the cohomology groups with coefficients in the corresponding local systems together with the induced linear maps between them?

As a side question, is there any hope of a similar interpretation of stale constructible sheaves in algebraic geometry with exit paths replaced by torsors of some kind?

• For your side question, I guess the canonical answer is this paper – Denis Nardin Apr 14 '20 at 13:54
• Have you looked in Lurie's draft book: Higher Algebra in Appendix A. math.ias.edu/~lurie/papers/HA.pdf and the various works that cite that source? – Tim Porter Apr 14 '20 at 15:57