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nLab > Latest Changes: enriched monad

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  1.  
    • CommentRowNumber1.
    • CommentAuthorSam Staton
    • CommentTimeAug 14th 2019

    expand with definition

    diff, v3, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2019

    So, for some examples, closure operators for 2\mathbf{2}, presumably. Anything interesting for metric spaces?

    • CommentRowNumber3.
    • CommentAuthorSam Staton
    • CommentTimeAug 15th 2019

    reference

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorSam Staton
    • CommentTimeAug 15th 2019
    • (edited Aug 15th 2019)
    Yes, closure operators would be an example, and I notice that the Kleisli presentation is indeed a known slick way of describing closure operators, which is quite nice. Not sure about the metric example, good question. What role do adjunctions play in the metric setting? Sorry if I have forgotten.

    Another example is that people often use cpo-enriched monads in CS for the semantics of iteration.
    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 15th 2019

    Added the example of closure operator, and linked to enriched adjunction.

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2019

    Is there a standard reference for the enriched version of things like Kleisli, Eilenberg-Moore categories, monads as lax 2-functors etc.?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 20th 2019

    Well, from one perspective there’s not much to say, since VCatV Cat is a 2-category and all of that is 2-categorical abstract nonsense. But I think there’s some particular discussion of enriched monads in Dubuc’s book Kan Extensions in Enriched Category Theory.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 20th 2019

    Thanks!

    diff, v7, current

  9. remove redundant ZZ in definition.

    Sandro Stucki

    diff, v9, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2023

    made explicit all the tacit assumptions in the definition

    While I was at it, I adjusted wording throughout,

    and gave some paragraphs to the Idea-section

    diff, v10, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2023
    • (edited Aug 22nd 2023)

    Incidentally, the references previously provided in the entry seem to have little relation to the current content of the entry. Or else it’s hard to spot.

    I have added pointer to:

    where the explicit definition of enriched monads in Kleisli form is Def. 5.6.

    (It’s not a priori clear that their definition – which is the obvious one – is equivalent to the one in the entry: The entry requires the structural equations to be satisfied on global elements only. Is this intentional?)

    [edit: Hm, since the equations appear in revision 3 without any typing, they are probably meant to apply to generalized elements. I’ll edit accordingly, but need to interrupt for the moment…]

    diff, v12, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2023

    have added the missing diagrams to the definition

    diff, v13, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2023

    expanded on the statement of the relation between enriched and strong monads and added a reference (here)

    Since this statement is being alluded to in various other entries, I am thinking of making this an !include section

    relation between strong and enriched functors and monads – section

    Not now, but maybe tomorrow…

    diff, v13, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2023

    added pointer to

    which has a detailed account of the relation enriched/strong/monoidal monads.

    diff, v16, current

1 to 14 of 14

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