If M is a paracompact manifold and P → M is a principal G-bundle, then there exists a map f : M → BG, unique up to homotopy, such that P is isomorphic to f ∗(EG), the pull-back of the G-bundle EG → BG by f.
Let G be a compact Lie group. There exists a contractible space EG on which G acts freely. The projection EG → BG is a G-principal fibre bundle.
So many decades since I did abstract nonsese. Sounds like the gist of it is the assignment of classifying spaces to (topological, lie, etc...) groups is not just a function but a functor, and a representable one at that.
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16SgJCRH7fB5ny6wCHabmiGaa6wcbve6WS Verified🙇♂️ precisely. logarithmic and exponential maps are what we see in Lie theory, though these maps can be far more general (specifically re: fibration).
it seems you can just continue to continue to group, map, repeat and continue to go higher
I'm not really actually good at homotopy theory. I'll be honest, as soon as diagrams started looking like hexagrams with all them H-spaces I started to die a little, and there’s a reason I never got a PhD.
But thing as you mentioned exponentiation, one thing I always thought was nifty was the way Quantum mechanics really lives in some category that’s more the like the exponentiation of Hilb. Where categorical product is tensor product. Mixed states.
who needs a PhD? i'm actually just diving into homotopy but it seems to be quite interesting for abstract definitions of continuous maps.
that's an interesting theory. i wonder what the logarithmiation (yes) of a Hilbert space would be in this case?