For a Lie group $G$, we can define a principal $G$ bundle as a submersion of manifolds $\pi:P \to X$ equipped with a free right $G$-action on $P$ that is transitive on the fibres over $X$.


What goes wrong with an analogous definition for 2-groups? For now, we can think of 2-groups as weak monoidal groupoid with monoidal inverses. The autoequivalences of a category $\operatorname{Aut}(C)$ form a 2-group. One can define a right action of a 2-group on a category $C$ as a monoidal functor from the 2-group to $\operatorname{Aut}(C)$. I will also consider only essentially finite two groups.


Now given a 2-group $G$, let $\pi:\mathfrak{P} \to \mathfrak{X}$ be a representable submersion of stacks over the category of manifolds (called $\mathrm{Man}$) equipped with the étale site. Suppose we define a "principal $G$ bundle" by an action of G on the category $\mathfrak{P}$ (on the right) such that the functor
\begin{gather\*}
 \mathfrak{P} \times G \to \mathfrak{P} \times\_{\mathfrak{X}} \mathfrak{P} \\
 (p,\gamma) \mapsto (p\cdot\gamma,p)
\end{gather\*}
is an equivalence of categories over $\mathrm{Man}$. Note that the action is over $\mathrm{Man}$, i.e. $p$ and $p\cdot \gamma$ are over the same object.


I could not find such a definition in the literature. Does something go wrong? If such a definition is available, please share a reference.



Categorifying the definition of a principal $G$ bundle
The definition you are looking for is precisely Def. 6.1.5 in:


*Nikolaus, Thomas; Waldorf, Konrad*, [**Four equivalent versions of nonabelian gerbes**](https://doi.org/10.2140/pjm.2013.264.355), Pac. J. Math. 264, No. 2, 355-420 (2013). [ZBL1286.55006](https://zbmath.org/?q=an:1286.55006).


This definition is based on the work of Toby Bartels ([Higher gauge theory I: 2-bundles](https://arxiv.org/abs/math/0410328)) and Christoph Wockel ([Principal 2-bundles and their gauge 2-groups](https://arxiv.org/abs/0803.3692)). The main advantage of the definition in the paper of Thomas Nikolaus and me is that it is as simple as possible: the action functor is a smooth functor (as opposed to a bibundle, or anafunctor), and the action is strict. Yet, principal 2-bundles in this sense are classified by Giraud's non-abelian cohomology, and are equivalent to bundle gerbes.


