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    • CommentRowNumber1.
    • CommentAuthorRichard Williamson
    • CommentTimeJan 19th 2020
    • (edited Jan 19th 2020)

    An attempt to create this page was made by Paulo Perrone, but the creation was not successful. Am creating the page without any content beyond ’TODO’ now as a test.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 19th 2020

    Okay. I’ve tried to insert the content, now it says “Edit blocked by spam detector”. I hope I’m not that bad at category theory… :p

  1. Adding Paulo’s content to circumvent spam filter.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 21st 2020

    Added idea section (more to come) - let’s see if my edits are accepted this time.

    diff, v4, current

    • CommentRowNumber5.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 21st 2020

    They worked. Thanks Richard!

    • CommentRowNumber6.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 21st 2020

    Added some examples

    diff, v5, current

    • CommentRowNumber7.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 21st 2020

    Added free algebras

    diff, v5, current

  2. hope it’s ok to add this link. It seemed weird not to have a link to that page, starting from this one.

    Nathaniel Virgo

    diff, v9, current

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 24th 2020

    Absolutely. But it might go better in a section on “Properties” or something, than in the “Definition” section.

    (BTW, it’s helpful to say in the changes box what you did, rather than making the forum reader click on the diff to figure out what “this link” refers to.)

    • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 30th 2021

    Added example of pointed sets being algebras for ()+1(-)+1.

    diff, v11, current

  3. Clarification that the full subcategory of free algebras is equivalent to the kleisli category.


    diff, v12, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeOct 22nd 2022

    I have touched wording and formatting in the section “Free TT-algebras” (here) in the hope to increase readability and usefulness.

    diff, v14, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 1st 2022

    I have tried to polish-up the section “Definition – Algebras” (here), touching its typesetting, formatting, and hyperlinking

    diff, v16, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2022
    • (edited Nov 2nd 2022)

    Just a question on the literature:

    On slide 4 in his Lecture 2, Uustalu states a way of speaking about algebras over monads that meshes with the Kleisli-style of presenting (strong) monads in FP, and refers to it as being “Kleisli triple algebras” or “Mendler-style algebras”.

    What would be a real reference for this “Mendler-style”-presentation of algebras-over-monads-in-FP, or anything along these lines?

    • CommentRowNumber15.
    • CommentAuthorvarkor
    • CommentTimeNov 2nd 2022

    What would be a real reference for this “Mendler-style”-presentation of algebras-over-monads-in-FP, or anything along these lines?

    There are two references that appeared in 2010: algebras for a relative monad (taking J = 1) in Monads need not be endofunctors, and algebras for extension systems in Monads as Extension Systems - No Iteration is Necessary (though the journal/arXiv version of the former paper is from 2015). An earlier reference is the algebras for a device in Walters’s An alternative approach to universal algebra, though it takes some decoding to see that this is really the same.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 3rd 2022


    I have now added these references to the entry (in a new subsection starting here)

    diff, v17, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 5th 2022

    Just a playful thought on terminology:

    1. I notice that the term “algebra over a monad” has some justification in classical universal algebra, but in many or most other situations, the term “module for a monad” seems more appropriate.

    2. If, however, ones says “module over a monad”, then it stands to reason that one might just as well refer to the monad as the monoid that it is and stick with the standard terminology of internal algebra.

    3. Conversely, if one does insist on “monad” (as we are bound to do) then it seems natural to make analogous suffix adjustments systematically to all related terms!

    4. Following this logic one will easily be led to say “modale” for “module over a monad”.

    5. But by lucky and fun conincidence, what starts out like a pun happens to make perfect sense with respect to thinking of monads as modalities: Their algebras are then the modal objects.

    So where a monoid has modules, we should say that a monad has modales :-)