Talk:Beck's monadicity theorem
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[edit]"Passing to a category of coalgebras for a comonad T is a high-flown way of modelling what taking equivalence classes does, in less touchy situations." What? a few more details, or a reference, would be useful.
128.135.60.45 03:56, 8 August 2007 (UTC)
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Compact Hausdorff spaces
[edit]I think the claim that that U preserves all colimits is wrong, see this question on mathoverflow.
The monadicity theorem does indeed prove that U is monadic though - but the proof is a little bit more involved and relies on assumption that the coequalizer is U-split. --Mz147 (talk) 18:47, 15 June 2021 (UTC)
I removed the compact Hausdorff space example, because it is definitely false that the forgetful functor preserves all colimits. (For example, let $X_n$ be the quotient of $[0,1]\times\{0,1\}$ in which $(x,0)\sim(x,1)$ for $x\geq 2^{-n}$; then the colimit in compact Hausdorff spaces is just $[0,1]$ but the colimit in spaces has two copies of $0$.) The standard proof of Manes' Theorem does not use Beck's Theorem. If there is an alternate proof using Beck's Theorem then it would be good to include that, but I do not know where to find one. Neil Strickland (talk) 16:47, 28 May 2022 (UTC)