Peck by twetch#7453

1NUNwL…2iDE Key · twetch

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.

twetch#7453 6.4y

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.

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Transaction: c6e83e5c51e0b679b9cd40ce33fc104640c2a59c347cc9527531844b493a78d7
Block: 634 004
Time: 2020-05-08T17:10:55Z
Signer: 1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE
App: twetch
Type: post
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Name in tx: twetch#7453

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Replies (6)

twetch#21877 6.4y

16SgJC…e6WS Key · twetch

Replying to@1NUNwL…2iDE

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.

twetch#7453 6.4y

1NUNwL…2iDE Key · twetch

Replying to@16SgJC…e6WS

🙇‍♂️ 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

twetch#7453 6.4y

1NUNwL…2iDE Key · twetch

Replying to@1NUNwL…2iDE

Reposted

twetch#7453 6.6y

twetch#21877 6.4y

16SgJC…e6WS Key · twetch

Replying to@1NUNwL…2iDE

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.

twetch#21877 6.4y

16SgJC…e6WS Key · twetch

Replying to@16SgJC…e6WS

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.

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