hmm what is the "best" way to think about the fact that H^1(-,G) classifies G-torsors? like in general??
u can write down cech cocycles for both and they give u the same thing so that's fine...
but like umm
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in Spaces I can say that H^1(X,G)=[X,BG] and then I am saying that the de-looping of G is the classifying space for G-torsors. And every G-torsor is a pullback of EG->BG.
Can I do the same thing inside a sheaf topos? Or a general topos???
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I guess one thing that I am feeling weird about is that here we are working inside the ∞-topos of Spaces over a point, and classifying objects over some obejct X in terms of maps out of X
but in the sheafy situation we are working in the ∞-(sheaf topos) over X...
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and still just classifying objects over X
I guess things will probably look nicer if I keep track of slice categories/work with families of torsors more systematically...
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if you work in the slice category of spaces over X then BG is the constant family X x BG (with the projection to X) and global sections of this "sheaf" (that is, the space of maps from the "point," which is the constant family X) is exactly the mapping space [X, BG]
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yeah I got that much... just was feeling kinda bothered by the seemingly different roles of the base object in each situation. But I think I am prolly happy with it for now...
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