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Gurkenglas (Jan 14 2022 at 14:40):An idempotent e=ee is split iff e=sf with fs=1. That sounds a lot like a monad e that arises from an adjunction f⊣s, where the counit corresponds to fs=1, but I don't see what would be the unit of the adjunction or the monad. What's going on?
Morgan Rogers (he/him) (Jan 14 2022 at 14:44):Have you encountered the nlab pages on [[idempotent monad]] and [[idempotent adjunction]]?
Gurkenglas (Jan 14 2022 at 14:52):The first yes, the second no, thanks. Note that f,s,e are not necessarily functors, just morphisms in some 1-category for which I may or may not find sensible 2-cells. (The Kleisli category of the double-dual monad.)
Morgan Rogers (he/him) (Jan 14 2022 at 14:56):The definition of idempotent adjunction lifts to the 2-categorical notion of adjoint 1-morphisms, if that's what you mean?
Gurkenglas (Jan 14 2022 at 15:02):It's good to know that nothing Cat-specific is going on. Can every split idempotent be seen as an idempotent adjunction?
Gurkenglas (Jan 14 2022 at 15:11):Presumably one can take the 1-category and freely add a 2-cell 1=>e for every idempotent e. Is that sensible? (And identify f(1=>sf) with id or equivalently (1=>sf)s with id.)
Mike Shulman (Jan 14 2022 at 15:38):Is that sensible?
In a word, no. (-:
I think a better way to explain the similarity is that both idempotent-splitting and the construction of the [[Eilenberg-Moore category]] of a monad are limits. The first is an ordinary 1-categorical limit, the second is a 2-categorical limit. (Idempotent-splitting is also a colimit; the corresponding 2-categorical colimit is the [[Kleisli category]].)
Asad Saeeduddin (Jan 26 2022 at 00:24):@Mike Shulman is there some sensible weakening of the notion of an adjunction so that a section/retraction pair in a category viewed as a trivial bicategory (2-morphisms being equalities) is precisely an instance of this notion?
Asad Saeeduddin (Jan 26 2022 at 00:40):it feels to me like a split idempotent is "half the consequences of an adjunction", in that it gives you a counit FG = 1 and a multiplication GFGF = GF
Asad Saeeduddin (Jan 26 2022 at 00:40):i'm not sure if there's simply a way to axiomatize that half of the consequences to end up with a generalization of the notion of an adjunction (i'm also not sure what the equations would be, but i hope we can make some up so that both halves put together would imply the zigzag identities)
Mike Shulman (Jan 26 2022 at 01:42):No, I don't think so.
Gurkenglas (Jan 27 2022 at 20:12):The relationship between adjunctions and "2-"retractions feels more like this https://sketchtoy.com/70400092