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I think thinking in terms of classifying spaces will help clarify the situation. We know that a rank $n$ complex vector bundle $V$ on $X$ is the same thing as a homotopy class of maps $f:X\to BU_n$. A …

What you are referring to are sometimes called stacks or sheaves of categories. A famously important example is the stack $\mathrm{QCoh}$ sending a scheme $U$ to the category of quasi-coherent sheaves …

Let's start approaching the question from the simplest possible case $Y=*$. What should be the points of $[X/G]$? Recall that the idea here is to generalize the construction of the action groupoid fo …

The first step is to construct a pairing on the modules $\bigwedge^k M$. I will assume that the pairing $g:M\otimes M\to R$ is perfect, that is it induces an isomorphism $M\to \textrm{Hom}(M,R)$. Then …

To complement Mark Grant's excellent answer, I'll say something more about the general case. This topic goes under the name of obstruction theory. The first observation is that a $G$-bundle on $X$ is …

My favourite reference for understanding spectral sequences is Boardman, J. Michael. "Conditionally convergent spectral sequences." Contemporary Mathematics 239 (1999): 49-84 (pdf). I don't think I …