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    • CommentRowNumber1.
    • CommentAuthorjcmckeown
    • CommentTimeAug 11th 2010
    a description of a free monad on Sets in Cat, the successor monad, and remarks.
    • CommentRowNumber2.
    • CommentAuthorjcmckeown
    • CommentTimeAug 11th 2010

    (that is, I'm not sure it's anywhere else called "the successor monad"; it's the monad for pointed sets in Set, I suppose... )

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeAug 12th 2010
    • (edited Aug 12th 2010)

    Handy link: monad

    I disagree that this is a monad on the (large) poset of all (material) sets under inclusion. The multiplication transformation μ\mu is not \subseteq; that is, SSXS S X is not a subset of XX.

    So we can only say that SS is

    • a monad on the category SetSet, with
    • a monic unit transformation ι\iota, such that
    • we can easily define it on material sets so that ι\iota is inclusion of subsets.

    I’ll rewrite this in a bit.

    PS: I like the name ‘successor monad’. I have found one previous usage (PostScript gzip), and it agrees with yours.

    • CommentRowNumber4.
    • CommentAuthorFinnLawler
    • CommentTimeAug 12th 2010

    Perhaps it should be on a page of its own?

    Also, your query is more or less answered at simplex category, or in Categories WorkΔ a\Delta_a is generated by the monoid object [0][1][2][0] \to [1] \leftarrow [2].

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeAug 12th 2010

    Perhaps it should be on a page of its own?

    Well, the list monad has a page of its own, so the successor monad could too. In the meantime, I’ve redirected it.

    your query is more or less answered […]

    I’ve put that in.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 12th 2010

    In the context of monadic functional programming, the successor monad is also called the maybe monad or the “error monad”. Here you’re thinking about the Kleisli category, which as pointed out in the example is equivalent to sets+partial functions – hence the idea that a Kleisli morphism is something that “maybe” gives you an output, or gives you an output unless it fails with an error.

    • CommentRowNumber7.
    • CommentAuthorjcmckeown
    • CommentTimeAug 12th 2010

    Well, thanks, all! I see there are now entries for Kleisli and Eilenberg-Moore. :-)

    Toby,

    I disagree … The multiplication transformation μ\mu is not \subseteq

    OK, I must have missed that seminar, or maybe should I read “Moore Closure”?

    Finn,

    Also, your query is more or less answered …

    While some will agree with the “more” bit, I’m feeling “less” enlightened than I perhaps should… let me muse on that a bit… monad == monoid made of a 11-cell in End(*)End(*) of a 00-cell in a 22-cat … maybe I’ll get it soon.

    • CommentRowNumber8.
    • CommentAuthorFinnLawler
    • CommentTimeAug 12th 2010

    If XX is an object of a monoidal category CC, then (X)(- \otimes X) is a monad if and only if XX is a monoid object. The restriction of SS to Δ a\Delta_a is ([1])(- \oplus [1]). So I think (but do correct me if I’m wrong) that the content of your observation is essentially that Δ a\Delta_a is generated by [1], as explained under Universal properties at simplex category, or CWM VII.5.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeAug 15th 2010

    The multiplication transformation μ\mu is not \subseteq

    OK, I must have missed that seminar, or maybe should I read “Moore Closure”?

    Well, you certainly could read Moore closure; I like to think that it’s a very nice article, since I wrote most of it!

    But that shouldn’t be necessary. You wrote that the monad was a monad on the poset of (presumably material) sets under inclusion, and in that case every relevant morphism has to be a morphism in this category, and μ X:SSXX\mu_X\colon S S X \to X is not. So that’s why I changed it to a monad on the category of sets and functions instead.

    By the way, your links will work better if you make them like [[Moore closure]] and [[Kleisli category]] instead of like [[Moore Closure]] and [[Kleisli Category]].

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