The path-groupoid $\mathcal{P}_1(X)$ of a (smooth) topological space $X$ is a refinement of the fundamental groupoid $\Pi_1(X)$ whose morphisms are given by (piecewise smooth) paths in $X$ up to thin-homotopy (rather than full homotopy). Thin-homotopies are basically homotopies sweeping zero area. (Note that there's a natural map $\mathcal{P}_1(X)\to \Pi_1(X)$ by sending thin-homotopies to full-homotopies.)

When we have a topological stratified space $(X,S)$, we may consider a different refinement of the fundamental groupoid $\Pi_1(X)$ given by the *exit-path category* $\text{Exit}(X,S)$. In particular, instead of considering the full set of paths, we look at the set of *exit-paths*: a path $\gamma:[0,1]\to X$ is an exit-path with respect to the stratification $S$ if, for each $t_1\leq t_2\in [0,1]$, the dimension of the stratum containing $\gamma(t_1)$ is less than or equal to the dimension of the stratum containing $t_2$. That is, exit-paths go up the strata. The morphisms of the exit-path category $\text{Exit}(X,S)$ are then given by exit-paths modulo (full) homotopy. (Note that $\Pi_1(X)\hookrightarrow EP(X,S)$ with respect to the trivial stratification.)

There are two sheaf-theoretic characterizations of the categories $\Pi_1(X)$, $\text{Exit}(X,S)$ given by corresponding equivalences of categories, namely:

the category of representations of the fundamental groupoid $\Pi_1(X)$ is equivalent to the category of local systems (say, for $X$ locally simply connected); see, for instance, section 2.6 Szamuely;

the category of representations of the exit-path category $\text{Exit}(X,S)$ is equivalent to the category of constructible sheaves; due to MacPherson, see Treumann; also mathoverflow.

### Question

Is there an analogous sheaf-theoretic characterization of the category of representations of the path-groupoid $\mathcal{P}_1(X)$?

transport functors. I am under the impression that these only give a particular sector of the category of representations of $\mathcal{P}_1(X)$ satisfying further descent data, therefore missing "all the others". Correct me, please, if I'm wrong. $\endgroup$4more comments