# Proof of MacPherson's result about set-valued constructible sheaves and exit paths

I'm looking for a proof of a theorem that is attributed to MacPherson. Treumann (Section 1.1 in Exit paths and constructible stacks, 2009) states the theorem as:

Theorem 1.2 (MacPherson). Let $(X,S)$ be a topologically stratified space. The category of $S$-constructible sheaves of sets is equivalent to the category $\text{Funct}(EP_{\leqslant 1}(X, S), \text{Sets})$ of $\text{Sets}$-valued functors on $EP_{\leqslant 1} (X, S)$.

Here $EP_{\leqslant 1}$ is the category of exit paths. He does not provide a proof, and earlier mentions

MacPherson (unpublished) observed that, for a fixed stratification $S$ of $X$, it is possible to give a description of the $S$-constructible sheaves on $X$ in terms of monodromy along certain paths.

So it seems the proof has not been published by MacPherson, but maybe someone else has written it down, and I would be very glad to see it. Barwick-Glasman-Haine (Introduction in Exodromy, 2018) similarly say

Bob MacPherson proved that constructible sheaves of sets on a (suitably nice) stratified [topological space] $X$ over a poset $P$ determine and are determined by a functor from the exit-path category.

No reference is given here. Ayala-Francis-Rozenblyum prove a similar version of MacPherson's theorem with functors into spaces (Corollary 3.3.11 in A stratified homotopy hypothesis, 2017), but they require $X$ to be a "conically smooth stratified space," which I would like to not impose. The most general case (I think) is given by Lurie (Theorem A.9.3 in Higher Algebra, 2017), which is my motivation here - in trying to understand Lurie's proof I am looking for a proof to MacPherson's statement.

Worth mentioning is that MacPherson considers actual paths and $\text{Set}$-valued functors, Treumann paths, 2-simplices and $\text{Cat}$-valued functors, and the rest use all dimensions and functors valued in spaces. I am specifically looking for a proof of the 1-dimensional $\text{Set}$-valued case (I understand that Treumann's and Lurie's proofs contain this case, but I guess I don't yet trust my own skills to pick out only the necessary pieces).

• This paper of Curry and Patel is pretty accessible and detailed, and focuses on the 1-categorical situation. It does restrict itself to the conically stratified case (which in most situations one cares about is satisfied anyway). – R. van Dobben de Bruyn Mar 17 at 18:56