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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJan 28th 2010

    Created idempotent adjunction. Does anyone know a published reference for this notion?

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJan 29th 2010

    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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJan 30th 2010

    Not sure what you mean. The terminology exists, I didn't invent it.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJan 31st 2010

    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.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2010

    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.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2010

    Thanks, then I have some major gap in the understanding the subject.

    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2010
    • (edited Feb 1st 2010)

    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 ?

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 1st 2010

    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.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeFeb 1st 2010

    Oh great, this explains what I overlooked! Neat.

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