Timeline for Can one compare monads arising from homotopy equivalent adjunctions?
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| Mar 19, 2013 at 7:15 | comment | added | Gregory Arone | Hi Peter. This is a helpful answer, thanks. It is interesting that the assumption that the map $L\to L'$ is an equivalence is not directly relevant to your argument. What matters is that $R$ and $R'$ can be moved past the bar constructions. The assumptions end up being equivalent to a form of Barr-Beck criterion for monadic descent. They guarantee that $T$ and $T'$-algebras embed into $\mathcal D$. You show that the images are the same. I had wondered whether the assumption that the maps $\alpha_L, \alpha_R$ are equivalences would give a more direct way to compare the monads. Perhaps not. | |
| Mar 19, 2013 at 3:45 | comment | added | Dylan Wilson | re sledgehammers: This is definitely laziness on my part. But I am somewhat okay with this... it's nice when the proof of some fact is as intuitive as the idea behind it. Now, both of our "proofs" (maybe mine's wrong...) had the same outline "prove it for free things, then do a little formalism to get everything else", we just went about it with slightly different language. | |
| Mar 19, 2013 at 3:40 | comment | added | Dylan Wilson | off the top of my head, now that I think about it.) Like you, I haven't had time to look at the multiplicativity statement. I could just be wrong! :) | |
| Mar 19, 2013 at 3:39 | comment | added | Dylan Wilson | Teasing is always welcomed. For the type of "uniqueness" I wanted, see the paragraph (and ensuing proofs if you like) beginning section 6.2.3 of Lurie's Higher Algebra. What I meant about the bar construction is just this: in the case given, algebras over a monad are generated by sifted (homotopy) colimits of free algebras. So if the given equivalence $T \rightarrow T'$ induces a map $T-alg \rightarrow T'-alg$ preserving sifted colimits then we can reduce to the free case. I should have made this assumption more explicit (I can't think of a counterexample to this property of an equivalence | |
| Mar 19, 2013 at 2:11 | history | answered | Peter May | CC BY-SA 3.0 |