Categorified analog of Collier–Lerman–Wolbert’s transport functor equivalence via the functor F

Ascertain whether the functor F from the category of quasi-principal G-bundles equipped with strict connections over a Lie groupoid X to the category of transport functors from the thin fundamental groupoid Π_thin(X) to the quotient category of G-torsors provides a categorified analog of Theorem 4.1 of Collier–Lerman–Wolbert (2016) when one imposes the smoothness condition that for each x in X_0, the restriction map T|_{Π_thin(X,x)}: Π_thin(X,x) → Aut(T(x)) is a map of diffeological spaces.

Background

The thesis develops a parallel transport functor T_{C,ω}: Π_thin(X) → overline{G−Tor} for quasi-principal Lie 2-group bundles over Lie groupoids, and proves a smoothness property analogous to the manifold case by endowing Π_thin(X) with a diffeological structure.

In Collier–Lerman–Wolbert (2016), Theorem 4.1 establishes an equivalence between principal G-bundles over differentiable stacks and smooth transport functors over those stacks. The authors define a functor F mapping quasi-principal G-bundles with strict connections to transport functors and note that a similar smoothness condition can be formulated in their setting.

However, they explicitly state that it remains inconclusive whether F yields a categorified analog of the equivalence in Theorem 4.1 when the smoothness condition is imposed in their framework.