This question is motivated by the recent paper An invitation to higher gauge theory by Baez and Huerta, and the 2007 paper Parallel Transport and Functors by Schreiber and Waldorf.

Let $M$ be a smooth, finite-dimensional manifold. A **lazy path** in $M$ is a smooth function $\gamma: [0,1]\to M$ such that all derivatives of $\gamma$ vanish at $0,1$. A **homotopy of lazy paths** $\gamma_0,\gamma_1: [0,1] \to M$ is a smooth function $\Gamma: [0,1]^2 \to M$ such that all derivatives of $\Gamma(s,t)$ vanish near $t=0,1$, and such that $\Gamma(s,t) = \gamma_s(t)$ for $s = 0,1$. A homotopy of lazy paths is **lazy** if additionally we have that for each $t$, all the $s$ derivatives of $\Gamma(s,t)$ vanish near $s = 0,1$. A homotopy (of possibly non-lazy paths) is **thin** if $\text{rank}\\, d\Gamma < 2$ everywhere. Note that (non-lazy) thin homotopies include all reparameterizations, and so any (possibly non-lazy) piecewise-smooth path is thinly homotopic to a lazy path. Note also that lazy paths concatenate smoothly, and the concatenation of lazy paths is associative up to thin homotopy. Note also that if $\gamma^{-1}(t) = \gamma(1-t)$, then the concatenation $\gamma^{-1}\gamma$ is thinly homotopic to a constant path. Note also that lazy thin homotopies concatenate, and so define an equivalence relation, and if two paths are thinly homotopic, then they are lazily-thinly homotopic. So define $\mathcal P^1(M)$ to be the set of all lazy thin homotopy equivalence classes of lazy paths in $M$. It is a groupoid with base $M$.

The idea is to consider $\mathcal P^1(M) \rightrightarrows M$ as not just a groupoid but an infinite-dimensional Lie groupoid. I think I understand the smooth structure on $\mathcal P^1(M)$: a **curve** in $\mathcal P^1(M)$ should be precisely a (non-thin) homotopy of lazy thin paths, up to thin homotopy. It's not entirely clear to me that this defines a smooth structure. But it probably works in some formalism.

But if I really want to think of $\mathcal P^1(M) \rightrightarrows M$ as a Lie groupoid, then I should treat $\mathcal P^1(M)$ not just as a smooth space, but actually as an (infinite-dimensional) manifold, and there are various things to check about the maps (the source and target maps should be surjective submersions, etc.). And it's not clear to me how to write down a smooth manifold structure on $\mathcal P^1(M)$.

Here's what I'd like. Given a point in $\mathcal P^1(M)$, I'd like a neighborhood of it and a "diffeomorphism" between that neighborhood and some (Fréchet, maybe?) vector space, and I'd like it to be clear that the gluings are smooth. I can make a start: it's clear that the space of lazy paths in a finite-dimensional vector space is a vector space, and that thin homotopies respect addition, so that $\mathcal P^1(\mathbb R^n)$ is a vector space. It's not clear to me how to put a topology on it, and it's not clear that I can approximate $\mathcal P^1(M)$ by chopping $M$ into trivializable pieces, take $\mathcal P^1$ of each piece, and try to glue back together — thin homotopies can take a path in one trivializable patch and make it wrap around $M$ in a complicated way, providing it wraps back, for example.

Hence the question:

What is the manifold structure on $\mathcal P^1(M)$?

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