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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.

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  • $\begingroup$ 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). $\endgroup$
    – user98602
    Nov 2 '15 at 21:15
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The notion of morphism you want to use depends on what you would like to do with this category (as a random example, we could be interested in studying the category of topological manifolds and proper maps, instead of just continuous ones).

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 $F(g\cdot x) = g\cdot F(x)$.

As an additional remark, often you take the Lie group to be acting on the right on the fibres, not on the left.

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