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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 $\mathcal{P}_1(X)$?

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    $\begingroup$ Is the guess maybe "bundles equipped with a (not necessarily flat) connection"? But I don't think I understand what a 'thin homotopy' is well enough to stand by that claim. $\endgroup$ – Dylan Wilson Sep 27 '17 at 19:29
  • $\begingroup$ Which category are you asking for representations in? Vect? If so, then Dylan has it right. Similarly if reps are in GTors for G a Lie group (this is work of Schreiber and Waldorf) $\endgroup$ – David Roberts Sep 27 '17 at 21:33
  • $\begingroup$ @DavidRoberts I was thinking about representations in Vect. But, for starters, Sets would do. I am indeed aware of Schreiber & Waldorf's work (link) where they explore the so-called 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. $\endgroup$ – Carlos Sep 28 '17 at 0:05
  • $\begingroup$ As @DylanWilson guessed quite well, these functors encode the information of the parallel transport on bundles with (not necessarily flat) connection. And in a way, we could say that the category of representations of the path-groupoid is equivalent to the category of bundles with (not necessarily flat) connection. $\endgroup$ – Carlos Sep 28 '17 at 0:06
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    $\begingroup$ In addition to the above-cited work of Schreiber and Waldorf, I would like to advertise arxiv.org/abs/1501.00967, which removes some technical assumptions of Schreiber and Waldorf (such as sitting instants) and proves a similar result. $\endgroup$ – Dmitri Pavlov Sep 28 '17 at 23:09

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