Motivation for my query:

It’s a well-known incontrovertible fact that there exists a bijection between the set of isomorphism class of
principal $G$ bundles over a pleasant topological house $X$ and the set $[X,B’G]$ of homotopy class of steady maps from $X$ to the classifying house $B’G$(utilizing the completely different notation than typical for comfort) of the Principal $G$ bundles.

Now let $X$ be a topological house and let $U=cup_{alpha in I} U_{alpha}$ be a overlaying of $X$. Now additionally it is well-known that the functor $phi:C(U) rightarrow BG$ from the Cech Groupoid $C(U)$ of the duvet $U$ of $X$ to the delooping groupoid $BG$ of the topological group $G$ will be thought-about as a principal $G$ bundle over the house $X$. (For instance see definition 3.2 in https://arxiv.org/pdf/1403.7185.pdf).

If we transfer one step greater, that may be a weak 2-functor from the Cech 2-groupoid $C^2(U)$ to the deoopolng $B^2G$ of a Weak 2 group $G$ (For definition of Cech 2 groupoid and delooping groupoid of weak 2 group please verify instance 2.20 and part 3.2 of https://arxiv.org/pdf/1403.7185.pdf and the definition of Weak 2-group is present in https://arxiv.org/abs/math/0307200 )

then we arrive on the definition of Principal 2-bundle over the house $X$ the place the construction 2-group is the weak 2 group $G$ (see definition 3.eight in https://arxiv.org/pdf/1403.7185.pdf) which I suppose can be equal to the native description of Christoph Wockel’s definition of Principal 2 bundles within the definition 1.eight in https://arxiv.org/pdf/0803.3692. ( Although I didn’t verify rigorously that they’re certainly identical)

Now motivated from the observations above ,

My query is the next:

(1) Is a weak 2 functor $F:C rightarrow B^2G$ from a class $C$ to the delooping groupoid $B^2G$ of a weak 2-group $G$ generally is a sensible choice of definition of Principal bundle over a class the place the construction group is the 2-group $G$?

Or

(2) To get an acceptable notion of native trivialisation of a principal bundle over a class we’ve to in some way appropriately outline the notion of Cech Gropoid $tilde{Ch}(U)$of a “cowl $U$ on the class $C$“ (could also be coming from some Grothendieck pretopology on Cat, the class of small classes) after which think about the functor $tilde{F}:tilde{C}h(U) rightarrow B^2G$ as a definition of regionally trivializable Principal 2-bundles over a class?

I couldn’t discover any literature the place a notion of regionally trivializable Principal bundle over a common class is explicitly talked about. So any suggestion of literature on this course can even be very useful.

Additionally I’m curious to find out about it is corresponding notion in greater classes and within the context of infinity class .

Thanks.