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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2021
    • (edited May 19th 2021)

    I have added (here) pointer to:

    (This edit prompted by discussion in another thread of the same name: here)

    diff, v6, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2023
    • (edited Sep 16th 2023)

    This entry deserves a more contentful statement of what “monad transformers” are meant to be.

    I guess people really mean something close to: pointed endofunctors on the category MndMnd of monads, hence:

    For each monad \mathcal{E}

    • a new monad \mathcal{E}'

    • equipped with a monad morphism t :t_{\mathcal{E}} \colon \mathcal{E} \to \mathcal{E}'

    (that’s what the Haskell pages seem to define)

    and (do people consider this?)

    • the condition that the transformations t t_{\mathcal{E}} are natural.

    [edit: Only now following the links in the above thread, I am reminded that Sergei Winitzki said just that here. So I’ll go ahead and finally make the edit here…]

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 16th 2023

    I have re-written the Idea-section and mentioned the conceptualization of monad ransformers as pointed endofunctors on MndMnd.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 17th 2023

    added pointer to what I gather are the original articles:

    diff, v10, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023

    added a section (here)

    making explicit the definition of monad transformers by Liang, Hudak & Jones 1995

    and then proving that it is equivalent to monad morphisms in the sense of Maranda 1966.

    That is, disregarding the naturality issue which Liang et al. seem to rather gloss over. What I am proving is that a natural transformation between monads satisfies their respect for the bind- operation iff it respects the join in the sense of Maranda.

    (The statement/proof is evident/immediate, but I haven’t seen it mentioned anywhere before.)

    diff, v12, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023

    Oh, I see now: Naturality of the transformation is already implied by its respect for return and bind. Will edit…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023

    expanded the proof (here) to derive the naturality clause

    diff, v13, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2023

    added pointer to:

    • Tom Schrijvers, Maciej Piróg, Nicolas Wu, Mauro Jaskelioff Monad transformers and modular algebraic effects: what binds them together, in: Haskell 2019: Proceedings of the 12th ACM SIGPLAN International Symposium on Haskell (2019) 98–113 [doi:10.1145/3331545.3342595]

    who cite Liang et al. broadly but then state the compatibility condition in terms of the join.

    diff, v14, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeOct 9th 2023

    added a remark (here) that the previous discussion establishes a covariant functor from monads to their Kleisli categories.

    This deserves to be expanded on further, but I have to run now to attend a seminar…

    diff, v18, current