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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2011
    • (edited Oct 7th 2011)

    I added more to idempotent monad, in particular fixing a mistake that had been on there a long time (on the associated idempotent monad). I had wanted to give an example that addresses Mike’s query box at the bottom, but before going further, I wanted to track down the reference of Joyal-Tierney, or perhaps have someone like Zoran fill in some material on classical descent theory for commutative algebras (he wrote an MO answer about this once) to illustrate the associated idempotent monad.

    Some of this (condition 2 in the proposition in the section on algebras) was written as a preparatory step for a to-be-written nLab article on Day’s reflection theorem for symmetric monoidal closed categories, which came up in email with Harry and Ross Street.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeOct 8th 2011

    Thanks! I fixed a bit of formatting, and changed some appearances of RLR L to MM which it seemed like they wanted to be.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2014
    • (edited Jan 14th 2014)

    I have made more explicit the statement that the Eilenberg-Moore category of an idempotent monad induced by a reflection reproduces the underlying reflective subcategory; at idempotent monad, at reflective subcategory and at Eilenberg-Moore category, pointing also to Borceux vol 2.

    • CommentRowNumber4.
    • CommentAuthorvarkor
    • CommentTimeAug 20th 2021

    Added a reference to idempotent adjunction.

    diff, v53, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2021
    • (edited Oct 4th 2021)

    added this pointer:

    diff, v55, current

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeJun 27th 2022

    Add a characterisation of idempotence in terms of distributivity.

    diff, v57, current

  1. Continued renaming monad ’M’ to ’T’ in subsection {#AlgebrasForAnIdempotentMonad}.

    Bertalan Pecsi

    diff, v60, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2023

    To the list of equivalent characterizations I have added (here) the one saying that

    Hom 𝒞(T(c),U(a)) Hom 𝒞(c,U(a)) (T(c)fU(a)) (cηT(c)fU(a)) \array{ Hom_{\mathcal{C}}\big( T(c) ,\, U(a) \big) & \overset{\sim}{\longrightarrow} & Hom_{\mathcal{C}}\big( c ,\, U(a) \big) \\ \big( T(c) \overset{f}{\longrightarrow} U(a) \big) &\mapsto& \big( c \overset{\eta}{\longrightarrow} T(c) \overset{f}{\longrightarrow} U(a) \big) }

    diff, v62, current

    • CommentRowNumber9.
    • CommentAuthorhugopaquet
    • CommentTimeNov 12th 2023

    I have added the fact that if an idempotent monad is strong with respect to a monoidal structure on the underlying category, then it is also a commutative monad.

    diff, v63, current

    • CommentRowNumber10.
    • CommentAuthorncfavier
    • CommentTimeMar 7th 2024

    Added an original result formalised in Agda: the monoidal functor induced by a strong idempotent monad is idempotent. I plan to create the page idempotent monoidal functor soon; the definition is quite natural and can be found in Programming contextual computations by Dominic Orchard.

    diff, v65, current

    • CommentRowNumber11.
    • CommentAuthorncfavier
    • CommentTimeApr 18th 2024

    Mentioned well-pointed endofunctors in the definition.

    diff, v66, current

  2. Added example: abelianization


    diff, v67, current

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeApr 28th 2024
    • John Frank Adams, Zbigniew Fiedorowicz, Localisation and Completion with an addendum on the use of Brown-Peterson homology in stable homotopy, arXiv:1012.5020, based on 1973 lectures by Adams

    diff, v68, current