What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)?

Question

Motivation for my question:

It is a well-known fact that there exists a bijection between the set of isomorphism class of principal $G$ bundles over a nice topological space $X$ and the set $[X,B'G]$ of homotopy class of continuous maps from $X$ to the classifying space $B'G$ (using the different notation than conventional for convenience) of the principal $G$-bundles.

Now let $X$ be a topological space and let $U=\bigcup_{\alpha \in I} U_{\alpha}$ be a covering of $X$. Now it is also well known that the functor $\phi:C(U) \rightarrow BG$ from the Čech Groupoid $C(U)$ of the cover $U$ of $X$ to the delooping groupoid $BG$ of the topological group $G$ can be considered as a principal $G$ bundle over the space $X$. (For example see definition 3.2 in https://arxiv.org/pdf/1403.7185.pdf).

If we move one step higher, that is a weak 2-functor from the Čech 2-groupoid $C^2(U)$ to the deloopoing $B^2G$ of a weak 2 group $G$ (For definition of Čech 2-groupoid and delooping groupoid of weak 2-group please check example 2.20 and section 3.2 of https://arxiv.org/pdf/1403.7185.pdf and the definition of weak 2-group is found in https://arxiv.org/abs/math/0307200 )

then we arrive at the definition of Principal 2-bundle over the space $X$ where the structure 2-group is the weak 2 group $G$ (see definition 3.8 in https://arxiv.org/pdf/1403.7185.pdf) which I guess will be equivalent to the local description of Christoph Wockel's definition of Principal 2 bundles in the definition 1.8 in https://arxiv.org/pdf/0803.3692. (Though I did not check rigorously that they are indeed same)

Now motivated from the observations above ,

My question is the following:

(1) Is a weak 2-functor $F:C \rightarrow B^2G$ from a category $C$(considered as a degenerate 2 category) to the delooping groupoid $B^2G$ of a weak 2-group $G$ can be a good choice of definition of principal bundle over a category where the structure group is the 2-group $G$?

Or

(2) To get an appropriate notion of local trivialisation of a principal bundle over a category we have to somehow appropriately define the notion of Čech 2-groupoid $\tilde{Ch}(U)$ of a "cover $U$ on the category $C$" (may be coming from some Grothendieck pretopology on Cat, the category of small categories) and then consider the 2- functor $\tilde{F}:\tilde{C}h(U) \rightarrow B^2G$ (where $\tilde{Ch}(U)$ is considered as a degenerate 2 category ) as a definition of locally trivializable Principal 2-bundles over a category?

I could not find any literature where a notion of locally trivializable principal bundle over a general category is explicitly mentioned. So any suggestion of literature in this direction will also be very helpful.

Also I am curious to know about its corresponding notion in higher categories and in the context of infinity category.

Thank you.

Answer 1

It's just a Kan fibration with all fibres principal homogeneous $G$-spaces. Take an $\infty$-category $C$ and a functor $C\to BG,$ where BG is the classifying groupoid of an $\infty$-group (a grouplike $E_1$-space). Pulling back the overcategory projection $EG=BG_{/\ast}\to BG,$ where $\ast$ is the unique object of $BG$, gets you the Kan fibration you wanted.

A Kan fibration over a 1-category is a biCartesian fibration in groupoids, and being $G$-principal is a condition on the fibres.

To see this is the right condition, let $c$ be an object of $C$. Then $c$ is classified by a functor $\ast\to C,$ and the fibre over $c$ must be the pullback of $EG$ to a point, which is now just $\Omega BG\simeq G$ by recognizing $BG$ as the delooping of $G$.