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Created idempotent adjunction. Does anyone know a published reference for this notion?
This terminology, while has advantages, worries me, as does not correspond to idempotent monad but rather to the intersection of conditions for the idempotent monad and idempotent comonad. Interested to see how this will develop.
Not sure what you mean. The terminology exists, I didn't invent it.
If I understood correctly: The monad constructed out of idempotent adjunction is itself idempotent but not necessarily the converse. The same for a comonad. But for converse one needs both conditions (on the monad and on the comonad) simultanaeoulsy. If one would like to have the terminology in complete correspondence with the properties of associated monads of comonads, then the first half could be called idempotent adjunction, second coidempotent adjunction and the strong case you quote as biidempotent adjunction.
No, the converse also holds. Either one of the induced monad or comonad being idempotent implies that the other also is and that the adjunction is idempotent.
Thanks, then I have some major gap in the understanding the subject.
I can't get it. look idempotent monad is equivalent to the counit of the adjunction to be isomorphism, i.e. we have a reflective subcategory. Idempotent comonad means that we have a coreflective subcategory (the other adjoint). The two are not the same unless we are in a Frobenius situation. What am I getting wrong ?
An adjunction being an idempotent monad (i.e. a reflection) is equivalent to the counit being an isomorphism and the adjunction being monadic. A non-monadic adjunction can still induce an idempotent monad, and when it does it also induces an idempotent comonad and is an idempotent adjunction.
Oh great, this explains what I overlooked! Neat.
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