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

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.

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 $$G$$.
• @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. – Harry Gindi May 17 at 15:28