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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It is observed by MacPherson (unpublished) that the category of $S$ constructible sheaves of sets on $X$ is equivalent to the category from functors from $Ex(X,S)$ to the category of sets.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?

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    $\begingroup$ For your side question, I guess the canonical answer is this paper $\endgroup$
    – Denis Nardin
    Commented Apr 14, 2020 at 13:54
  • $\begingroup$ 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? $\endgroup$
    – Tim Porter
    Commented Apr 14, 2020 at 15:57
  • $\begingroup$ For $X \to P$ a stratified space that satisfies the exodromy equivalence for constructible $\infty$-sheaves, the terminal map $X \to *$ induces the terminal map $\operatorname{Sing}^P(X) \to *$ on exit-path $\infty$-categories, which again induces, by left and right Kan extension, an adjoint triple between the categories of constructible sheaves on $X$ and on $*$. The latter is however just the coefficient category. I would claim that the left Kan extension $C_*$ agrees with homology with coefficients in the constructible sheaf and the right Kan extension $C^*$ with cohomology. $\endgroup$
    – Markus Zetto
    Commented Jan 2, 2023 at 13:27
  • $\begingroup$ See math.ias.edu/~lurie/287xnotes/Lecture26.pdf for the non-stratified analogue (use the Dold-Kan correspondence); e.g. the right adjoint is similar to a (automatically derived) global sections functor. Also, note how this construction works for every stratified map, not only the terminal one. Applying Verdier duality further yields versions of Borel-Moore homology and cptly. supp. cohomology. Finally, your idea about gluing from local systems sounds very reasonable on the level of cochains; I'll try to figure out a precise statement. $\endgroup$
    – Markus Zetto
    Commented Jan 2, 2023 at 13:38

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