# Category of representations of the path-groupoid

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

• 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. Sep 27, 2017 at 19:29
• 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) Sep 27, 2017 at 21:33
• @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. Sep 28, 2017 at 0:05
• 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. Sep 28, 2017 at 0:06
• 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. Sep 28, 2017 at 23:09