# Category of $G$-bundles of class $C^k$?

Let $k\in \mathbb N\cup\{0, +\infty, \omega\}$, $G$ a $C^k$ Lie group and let $E$ and $M$ two $C^k$-manifolds. We say a $C^k$-map $\pi:E\longrightarrow M$ is a $G$-bundle of class $C^k$ if there is a $C^k$-manifold $F$ such that:

$(i)$ There is a free $C^k$-action $G\times F\longrightarrow F$, $(g, x)\longmapsto g\cdot x$;

$(ii)$ There is an open covering $(U_\alpha)_{\alpha\in I}$ of $M$ and a family of $C^k$-difeomorphisms $$(\phi_\alpha:\pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times F)_{\alpha\in I},$$ such that $$\textrm{pr}_1\circ \phi_\alpha=\pi|_{\pi^{-1}(U_\alpha)},\quad \forall \alpha\in I,$$ and the maps $$\phi_\alpha\circ \phi_\beta^{-1}:(U_\alpha\cap U_\beta)\times F\longrightarrow (U_\alpha\cap U_\beta)\times F,$$ are of the form $$(p, x)\longmapsto (p, g_{\alpha\beta}(p)\cdot x),$$ for a $C^k$-map $g_{\alpha\beta}:U_\alpha\cap U_\beta\longrightarrow G$.

Let us denote the bundle above by $G(E, \pi, M, F)$.

How can I define a category whose objects will be $G$-bundles of class $C^k$? More precisely, how do I define a morphism between two $G$-bundles of class $C^k$? Furthermore, is there a standard notation to this category?

Thanks.

• Usually one is only interested in $C^0$ or $C^\infty$. There is a canonical bijection on isomorphism classes of $C^\infty$-bundles and $C^0$-bundles over a smooth manifold, roughly because you can smooth maps (so smooth the map to the classifying space $BG$ in the appropriate sense).
– user98602
Nov 2 '15 at 21:15

The most natural kind of morphism you could define are the $G$-equivariant fibre bundle maps of class $C^k$, that is couples $(f,F)$ of $C^k$-maps with $f:M\to M'$, $F:E\to E'$ such that $f\circ \pi = \pi'\circ F$ and $F(g\cdot x) = g\cdot F(x)$.