References on principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is a category?

Question

Is there any treatment on principal "categorical" bundles - principal $\mathbf{C}$-bundles, where $\mathbf{C}$ is some (topological) category?

I know that one can define "categorical covering spaces" - locally constant $\mathbf{C}$-valued sheaves on a topological space $X$. I think that there's a nice equivalence of categories of the form

$$\left\{\text{Locally constant } \mathbf{C}-\text{sheaves on }X \right\} \leftrightarrow [\Pi_1(X), \mathbf{C}]$$

where $\Pi_1(X)$ is the fundamental groupoid of $X$ and $[\_,\_]$ is the functor category.

Is there a general notion of principal $\mathbf{C}-$bundle that captures this particular case? Is there any hope of a nice result in the form of a bijection

$$\left\{ \text{Principal } \mathbf{C}-\text{bundles on } X \text{ up to iso} \right\} \leftrightarrow [X, B \mathbf{C}]$$

where $[X, B\mathbf{C}]$ represents homotopy classes of maps from $X$ to the classifying space of the category $\mathbf{C}$?

Answer 1

Such a notion of a principal $\def\bC{{\bf C}}\bC$-bundle (when $\bC$ is a topological or simplicial category, or a Segal space) is available in Definition 6.1 of the paper Classifying spaces of infinity-sheaves.

It works as expected: the cocycle data of a $\bC$-bundle on $X$ is given by an open cover $\{U_i\}_{i∈I}$ of $X$ together with maps $U_i→{\bf Ob}(\bC)$, $U_i∩U_j→{\bf Mor}(\bC)$, etc., which must satisfy the appropriate higher coherence conditions, expressed via the Čech nerve. Some care must be exercised since $B\bC$ inverts all morphisms in $\bC$, which can be taken into account by modifying the Čech nerve appropriately.

Then one can define the space of $\bC$-bundles over $X$ (Definition 6.4) and prove that this space is a representable ∞-sheaf with respect to $X$. When $X$ is a manifold, this is proved in Theorem 6.6 of op. cit. When $X$ is an arbitrary topological space, one can instead cite Theorem 8.5 of Numerable open covers and representability of topological stacks, where open covers used to define $\bC$-bundles must be numerable.

Answer 2

You may find the following paper relevant, by thinking of your topological groupoid as a topological $2$-category with a single object (and invertible $2$-morphisms). The geometric nerve discussed there classifies the corresponding principal $2$-bundles.

Baas, Nils A.; Bökstedt, Marcel; Kro, Tore August
Two-categorical bundles and their classifying spaces.
J. K-Theory 10 (2012), no. 2, 299–369.

https://arxiv.org/abs/math/0612549

It generalizes

Baas, Nils A.; Dundas, Bjørn Ian; Rognes, John
Two-vector bundles and forms of elliptic cohomology.
Topology, geometry and quantum field theory, 18–45,
London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, 2004.

https://arxiv.org/abs/math/0306027

In the presence of a symmetric monoidal structure that you might want to group complete with respect to, the following two papers may be relevant:

Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit; Rognes, John
Stable bundles over rig categories.
J. Topol. 4 (2011), no. 3, 623–640.

https://arxiv.org/abs/0909.1742

Baas, Nils A.; Dundas, Bjørn Ian; Richter, Birgit; Rognes, John
Ring completion of rig categories.
J. Reine Angew. Math. 674 (2013), 43–80.

https://arxiv.org/abs/0706.0531

(Apologies for the self-promotion.)