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nLab > Latest Changes: relative monad

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    • CommentRowNumber1.
    • CommentAuthormattecapu
    • CommentTimeAug 3rd 2022

    Definition and characterization as monoids

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorSam Staton
    • CommentTimeAug 5th 2022

    mention when the skew monoidal structure is an ordinary monoidal structure

    diff, v7, current

    • CommentRowNumber3.
    • CommentAuthorSam Staton
    • CommentTimeAug 5th 2022

    Examples. Hope I got the monads with arities one right.

    diff, v7, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 5th 2022

    Put in the spaces, so it renders properly, ’J X’ rather than ’JX’, yields JXJ X rather than JXJX.

    diff, v8, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2022

    I have spelled out (here) in a fairly elementary fashion the elementary example of “forming linear spans”, namely of sending sets to the 𝕂\mathbb{K}-vector spaces which they span.

    In the course of doing so, I have adjusted wording and formatting in the Definition section, specifically in what is now the subsection “Definition – As skew-Kleisli triples” (here), in the hope to improve readability.

    (The Idea-section of this entry still needs attention, but I’ll leave it as is for the time being.)

    diff, v13, current

    • CommentRowNumber6.
    • CommentAuthormaxsnew
    • CommentTimeNov 17th 2022

    Add a less technical idea section, and add a section that shows JJ can be generalized to a profunctor.

    I hope no one minds I deleted some very technical material from the idea section that is immediately repeated in the Definition section anyway.

    diff, v14, current

    • CommentRowNumber7.
    • CommentAuthorvarkor
    • CommentTimeNov 17th 2022

    @maxsnew: I’m curious whether you have some motivating examples for this generalisation?

    • CommentRowNumber8.
    • CommentAuthormaxsnew
    • CommentTimeNov 18th 2022

    I gave a talk about relative monads in CBPV recently, and I have some programming examples there: https://www.youtube.com/watch?v=ooj1vJRixEU&list=PLyrlk8Xaylp5hkSMipssQf3QKnj6Nrjg_

    To summarize, in CBPV a monad relative to F : val -> comp is a more low-level version of a monad that specifies the stack the computation runs against. It’s natural to consider CBPV where F doesn’t necessarily exist and you can still define relative monads as relative to the “profunctor of computations” which is always present. Additionally, the morphisms of comp (the stacks) are typically not internalized as a data type in CBPV, but the elements of the profunctor are, so even if you have F, the notion of self-enriched relative monad needs to use the profunctor generalization.

    I thought it would be a bit too far afield to try to explain those examples on this page.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 18th 2022

    I find that a real-world example is just what this entry needs.

    Last night I have started watching the talk you pointed to (the one here). It’s nice, but at some point I admittedly missed how the crawling through the stack looking for exceptions is a relative monad. If that could be explained in the entry, it would be interesting.

    • CommentRowNumber10.
    • CommentAuthorJ-B Vienney
    • CommentTimeNov 18th 2022
    • (edited Nov 18th 2022)

    Maybe could you explain those examples in a new entry “relative monads in CBPV” and link it to the pages CBPV and relative monad?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeMar 23rd 2023

    added missing typing of “kk” in the associativity clause

    diff, v18, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2023

    added pointer to:

    diff, v19, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2023

    added reference pointer for the first class of Examples (here)

    diff, v19, current

    • CommentRowNumber14.
    • CommentAuthorBryceClarke
    • CommentTimeMay 28th 2023

    Added a few more recent references. There are many more to add.

    diff, v20, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2023

    Just for completeness, I have spelled out the example (here) of relative monads by precomposition of an actual monad with any functor.

    diff, v22, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 29th 2023

    added pointer to:

    diff, v24, current

    • CommentRowNumber17.
    • CommentAuthorvarkor
    • CommentTimeApr 6th 2024

    Updated the notation to match that of relative adjunction, cleaned up parts of the entry, and mentioned that relative monads are monoids in skew-multicategories.

    diff, v27, current

    • CommentRowNumber18.
    • CommentAuthorvarkor
    • CommentTimeApr 6th 2024

    Mentioned that relative monads are monads in the bicategory of distributors when JJ is dense.

    diff, v27, current

    • CommentRowNumber19.
    • CommentAuthorRodMcGuire
    • CommentTimeMay 12th 2024

    added

    doi:10.1016/j.jpaa.2024.107676

    to “The formal theory of relative monads”

    diff, v30, current

1 to 19 of 19

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