Since the notion of a graph of groups relies mostly on the pushout, can we construct graphs of objects in some other category, say, vector bundles? If this is the case and we have a "fundamental bundle" (analogous to the fundamental group of the graph of groups, a colimit of vertex bundles(?)) would this yield any extra information about the vector bundle?
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$\begingroup$ Example: Given a manifold $M$, a vector bundle $E$ (or sheaf), and an open cover $(U_\alpha )$, one has the $\check{C}$ech complex associated to the cover. Each simplex has a bundle given by the restriction of the original bundle $E$ to the intersection corresponding to that simplex. I might have to $\check{C}$ech the details a little more, but I think this is an example. $\endgroup$– ZxJxCommented Oct 9, 2019 at 20:13
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