For a Lie group G, we can define a principal G bundle by 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 Aut(C) forms 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 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 Man) equipped with the etale 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 $$\mathfrak{P} \times G \to \mathfrak{P} \times_{\mathfrak{X}} \mathfrak{P}$$ $$(p,\gamma) \mapsto (p\cdot\gamma,p)$$ is an equivalence of categories over Man. Note that the action is over 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.

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.

For a Lie group G, we can define a principal G bundle by 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 Aut(C) forms 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 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 equipped with the etale 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 $$\mathfrak{P} \times G \to \mathfrak{P} \times_{\mathfrak{X}} \mathfrak{P}$$ $$(p,\gamma) \mapsto (p\cdot\gamma,p)$$ is an equivalence of categories.

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

For a Lie group G, we can define a principal G bundle by 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 Aut(C) forms 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 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 Man) equipped with the etale 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 $$\mathfrak{P} \times G \to \mathfrak{P} \times_{\mathfrak{X}} \mathfrak{P}$$ $$(p,\gamma) \mapsto (p\cdot\gamma,p)$$ is an equivalence of categories over Man. Note that the action is over 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.