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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.

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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 $G$.

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    $\begingroup$ In complete generality, being a principal G-space is an additional structure that has to be specified and checked to be compatible with the data in the fibration, but this is the rough gist of it. $\endgroup$ – Harry Gindi May 17 at 9:41
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    $\begingroup$ Local triviality doesn't really make sense homotopically/categorically. It's not a fully homotopy-invariant property. The fact that this works for topological spaces is because you are working with nice spaces that have a close relationship between their topological and homotopical structure, somehow. $\endgroup$ – Harry Gindi May 17 at 9:51
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    $\begingroup$ @AdittyaChaudhuri The fundamental groupoid can only see the locally constant sheaves on your space. It's a special property of locally constant sheaves of sets that they can be reconstructed up to isomorphism from their homotopical data assuming that your original space is nice enough. $\endgroup$ – Harry Gindi May 17 at 15:00
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    $\begingroup$ @AdittyaChaudhuri The property is just the property of being a Kan fibration. $\endgroup$ – Harry Gindi May 17 at 15:08
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    $\begingroup$ @AdittyaChaudhuri Those are internal categories in $\mathbf{Diff}_\infty$, so they do carry a topology. These are much more complicated structures than what you are contemplating. $\endgroup$ – Harry Gindi May 17 at 15:28

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