Timeline for Why are monadicity and descent related?
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| Apr 4, 2022 at 12:38 | comment | added | Nikio | @QiaochuYuan Sorry to bother after so long the question was answered, but I'm very interested in some kind of reference or explaination for how the category of coalgebras over the comonad $f^*f_*$ is equivalent to (the category of descent data, and this is Benabou-Roubaud theorem I guess, and to) the category of homotopy fixed points. What defines the isomorphisms $A\to g^*A$ satisfying the cocycle condition once I have given a coalgebra $A \to f^*f_*A$ (and viceversa)? Thank you so much :) | |
| Sep 23, 2016 at 20:59 | comment | added | Qiaochu Yuan | @Heinrich: sure, and homotopy fixed points are not just special points, they also come with new structure. This is a standard feature of categorification; I don't think it invalidates the analogy, although I agree that idempotent monads are much more like idempotents than monads are. | |
| Sep 23, 2016 at 13:45 | comment | added | HeinrichD | I am not sure if I would agree with "monads are idempotents". Algebras for a monad are not just special objects, they come with new structure. At least idempotent monads behave like idempotents. | |
| Dec 16, 2015 at 16:56 | comment | added | Qiaochu Yuan | @Emily: yeah, I ran into this issue while writing a blog post fleshing this answer out (qchu.wordpress.com/2015/12/15/monads-are-idempotents) and I don't have a principled answer. The two agree if $m$ is an idempotent monad, so I guess the "monads are idempotents" intuition doesn't have enough resolution to distinguish the Eilenberg-Moore and Kleisli categories. | |
| Dec 16, 2015 at 15:35 | comment | added | Emily Riehl | "Taking the category of algebras" is a limit construction, more precisely analogous to "fixed points", but is there any reason to prefer this to the dual colimit construction, which yields the Kleisli category? | |
| Dec 16, 2015 at 15:32 | comment | added | Emily Riehl | In your analogy "monads are categorified idempotents" you say "the analogue of taking the fixed points of an idempotent is taking the category of algebras of a monad." Here "taking fixed points" means splitting the idempotent, forming the equalizer of $m, 1_X \colon X \rightrightarrows X$, but the same object is recovered by instead taking the coequalizer of this pair (because, as you note, a splitting is a "direct summand": both a submodule and a quotient)... [to be continued] | |
| Dec 7, 2015 at 14:18 | comment | added | Arrow | I have accepted Jon Beardsley's answer because it clarifies the link between algebraic structure and descent for rings. Still, your answer is very illuminating and I will surely ask more about it in time. | |
| Dec 6, 2015 at 19:57 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Dec 6, 2015 at 18:59 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Dec 6, 2015 at 18:53 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Dec 6, 2015 at 17:40 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |