Timeline for EM functor from monads to adjunctions
Current License: CC BY-SA 4.0
7 events
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| Apr 24, 2022 at 17:24 | comment | added | Tim Campion | That is, you didn't spell out what you mean by a morphism of adjunctions, but if you just take the functor 2-category $2Fun(Adj, Cat)$, you'll get something rather restrictive, since the 1-cells will be required to commute with both the left and right adjoints. You may have to allow more 1-morphisms in this category. | |
| Apr 24, 2022 at 17:21 | comment | added | Tim Campion | If you haven't already, you should look at Street's Formal Theory of Monads, which should clarify the situation. He explains how $EM : Mnd \to Cat$ is a 2-functor, right 2-adjoint to the 2-functor $i : Cat \to Mnd$ sending a category to the identity monad on it. The action of this 2-functor on 1-morphisms is what you're looking for, I think, even though the setup is a little different. BTW I'm not sure there really is a functor between Adj and Mnd given by EM -- I think on 1-cells you can get something which commutes with one direction of adjoints but only laxly commutes with the other. | |
| Apr 15, 2022 at 17:58 | comment | added | varkor | You may also be interested in this recent talk, which presents a novel perspective on the structure–semantics adjunction which has similar inspiration to your questions. | |
| Apr 13, 2022 at 17:39 | comment | added | varkor | Pumplün's article is freely available on a certain repository of scientific papers :) | |
| Apr 13, 2022 at 17:38 | comment | added | varkor | Unfortunately I still don't have time to expand, but Auderset's Adjonctions et monades au niveau des 2-catégories gives a fairly explicit derivation of the Eilenberg–Moore and Kleisli constructions from the 2-categorical perspective. Pumplün's is a different (albeit related) construction, but more relevant to the structure–semantics adjunction, because it also characterises the functors between Eilenberg–Moore categories and between Kleisli categories appropriately, which Auderset's does not. | |
| Apr 13, 2022 at 17:16 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 13, 2022 at 17:07 | history | asked | Alec Rhea | CC BY-SA 4.0 |