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nLab > Latest Changes: co-Kleisli category

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2015

    started co-Kleisli category with a minimum of content. Even though its formally dual to Kleisli category, of course, it may be worthwhile to have a separate entry.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 13th 2015

    It might also be known as the Kleisli category for a comonad (I should check e.g. Lambek and Phil Scott’s book). Some time back I wrote up an application of this concept to functional completeness.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2015

    Blute, Cockett and Seely say coKleisli in Differential categories.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2015

    I have added more redirects.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeMay 13th 2015

    I’m pretty sure I’ve seen “Kleisli category for a comonad”.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2015

    Google score 4 versus 1020 in favour of coKleisli category.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 13th 2015

    Confirming the usage in #2 here and here (result 6 of 11) and here.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 13th 2015

    I do think “coKleisli” is better, even though the other has been used; e.g. we say “category of algebras for a monad” and “category of coalgebras for a comonad” for the dual concepts.

    In principle, “opKleisli” might be even better, since both Kleisli and coKleisli are colimit constructions in CatCat; but probably that isn’t going anywhere. (-:

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 14th 2015

    I don’t think “coKleisli” or “co-Kleisli” is bad, even though I don’t think we say “co-Eilenberg-Moore” for the category of coalgebras (i.e., I think I’ve heard “Eilenberg-Moore category for a comonad”). Anyway, hopefully we have all the redirects we need.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 18th 2015

    an elementary question on terminology, that will just demonstrate my ignorance:

    for a left exact comonad JJ, then every Kleisli morphism from EE to FF canonically induces a function from the global points of EE to those of FF. What is the name of this construction?

    For definitenes, I mean the following map: Since JJ preserves the terminal object, then the image under JJ of a global point *→E\ast \to E is a global point *→J(E)\ast \to J(E). Postcomposing this with the given Kleisli morphism J(E)→FJ(E)\to F gives a global point of FF.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMay 19th 2015

    I don’t recall hearing of a name for that.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2015

    Okay, thanks.

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