Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topological topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 is not ; that is, SSX is not a subset of X.

    So we can only say that S is

    • a monad on the category Set, with
    • a monic unit transformation ι, such that
    • we can easily define it on material sets so that ι 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 WorkΔa is generated by the monoid object [0][1][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 is not

    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 1-cell in End(*) of a 0-cell in a 2-cat … maybe I’ll get it soon.

    • CommentRowNumber8.
    • CommentAuthorFinnLawler
    • CommentTimeAug 12th 2010

    If X is an object of a monoidal category C, then (X) is a monad if and only if X is a monoid object. The restriction of S to Δa is ([1]). So I think (but do correct me if I’m wrong) that the content of your observation is essentially that Δa 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 is not

    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 μX:SSXX 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]].