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nLab > Latest Changes: superconvex space

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  1. Working on my first new page in progress.

    Kirk Sturtz

    v1, current

  2. Adding fundamental properties and examples.

    Kirk Sturtz

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 20th 2022
    • (edited Jul 20th 2022)

    I have changed the first line

    Super convex spaces generalize the idea of a convex space by replacing finite affine sums with countable affine sums.

    to:

    The notion of “super convex spaces” generalizes the idea of convex spaces by replacing finite affine sums with countable affine sums.

    diff, v6, current

  4. (1) Added reference to Mackey. (2) Replaced useless example for ideals of N with a meaningful result involving Ideals showing why they are important. (3) Added remark under properties that the infinite-dimensional simplex is dense.

    Kirk Sturtz

    diff, v7, current

  5. Added Section for Related concepts to include links to {convex spaces, totally convex space}

    Kirk Sturtz

    diff, v11, current

  6. Made remark that the second axiom implies the first axiom.

    Kirk Sturtz

    diff, v13, current

  7. Added example of the probability monad on compact Hausdorff spaces to show how algebras (for a probability monad) can be viewed as a super convex space.

    Kirk Sturtz

    diff, v14, current

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeJul 27th 2022

    This is nice. But may I suggest adding the ’Idea’ sentence that is here on the nLab to someone in the intro (or even abstract) of your recent article? Super convex spaces are discussed but not defined, and then suddenly defined with no intuition given. I was under the impression that the ’super’ was meant in the sense of ’supergeometry’ or ’superalgebra’, when it’s not at all!

    • CommentRowNumber9.
    • CommentAuthorGuest
    • CommentTimeJul 27th 2022
    Thanks for the recommendation David. I will do that. (I never thought about that interpretation!) I am rewriting various aspects because at the n-Category Cafe Ruben van-Belle brought up a question which went straight to the heart of simplifying everything by cleaning up a small error. (Not necessary to drop down to the adequate subcategory of SCvx.)

    kirk

  10. Completed example 6.5 on quotient spaces and probing spaces via maps into the two objects NN and Δ N\Delta_{N} to show an important application for super convex spaces.

    Kirk Sturtz

    diff, v16, current

  11. Added an explanation for the alternative definition which is not so obvious without pointing out Lemma 5.1.

    Kirk Sturtz

    diff, v17, current

  12. Replaced Example 6.5 with a better explanation. This Isbell duality viewpoint explains how to get measurable spaces from super convex spaces.

    Kirk Sturtz

    diff, v18, current

  13. Added reference to Borger/Kemper, modified the alternative definition to its’ equivalent hom functor SCvx(⋅,A)SCvx(\cdot,A) definition because that is more useful for applications. In particular, for example 6.5, the hom functors are extremely useful.

    Kirk Sturtz

    diff, v23, current

  14. The last example has been completed with the general argument to prove the claimed adjunction. Also removed the remark on the arXiv reference article (by me) as it is now updated.

    Kirk Sturtz

    diff, v25, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 8th 2022
    • (edited Aug 8th 2022)

    Chasing references, I see that the term was introduced in

    (which I have added now) – but Pumplün (and Börger & Kemper following him) say superconvex space instead of super convex space.

    That makes a crucial difference and clarifies the issue raised in #8!

    I’d urge to rename the entry from “super convex space” to “superconvex space”.

    diff, v26, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 11th 2022
    • (edited Aug 11th 2022)

    following up on #15, I have replaced all occurrences of “super convex space” with “superconvex space” and have renamed the entry according

    diff, v27, current

  17. Per my discussion on MO concerning Idempotent Markov kernels with Fritz I have altered the last example so as to explicitly clarify what is meant by a free space and why superconvex spaces can be viewed as quotients of free spaces.

    Kirk Sturtz

    diff, v28, current

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 16th 2022

    A link to the MO discussion. I haven’t dug into the details of it, but there’s an unfinished disagreement going on in the comments with Fritz.

  19. I added the elementary lemma Every quotient of a free space is the free space of a quotient in the Example on free spaces. This simple result explains ALOT about the category, and the MO discussion I had with Fritz. Fritz was using the fact that the coequalizer of the idempotent Markov kernel gives another Markov kernel. Well the idempotent splitting of that kernel kk can be viewed as a map in the E.M. category, so taking the coequalizer of it and the identity map is trivial in E.M. But I was unaware of the elementary lemma at the time so it led to confusion on my part. Note the space 𝒢X\mathcal{G}X, viewed as a superconvex space, is a free space. Hence 𝒢X/ℛ≅𝒢(X/𝒮)\mathcal{G}X/ \mathcal{R} \cong \mathcal{G}(X/\mathcal{S}). I love that question because it explains the interplay between the Kleisi category and E.M. category (Superconvex space category), and why Fritz can do quite a fair amount of useful theory never leaving that category.

    Kirk Sturtz

    diff, v29, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 17th 2022
    • (edited Aug 17th 2022)

    I haven’t been following either, but maybe you could ask Tobias Fritz to come over here and look at the entry, to see if he agrees now? (He passes by here every now and then but maybe hasn’t seen your material yet).

    By the way, just to friendly tease you where you write, in capitals no less:

    This simple result explains ALOT

    I am being reminded of Mike’s quip here.

    • CommentRowNumber21.
    • CommentAuthorkirksturtz
    • CommentTimeAug 17th 2022
    I am sure Fritz has taken a gander at the material. I put a link to it, and he has independently figured out the same result, at least for the case $X$ finite.
    He is a proponent of using the Kleisi category (Markov cat) and I'm obsessive compulsive with EM, hence our discussions may appear like a disagreement but
    really we're just trying to get to the bottom of what's the best way to categorically tackle probability theory.

    Urs, never fear, I am doing my best to recruit people far more competent than me to check my gibberish. I'm just a member of the glee club for superconvex spaces.

  22. Feedback from Tobias Fritz on the Lemma I added yesterday concerning free spaces revealed I forgot to state a major hypothesis. That hypothesis has been added.

    Kirk Sturtz

    diff, v30, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2022

    Regarding the glee club, allow me to ask, out of my complete ignorance: Could you say what is the glee about superconvex spaces? Why do you and others care about them?

    (I am not doubting their use, but from looking at your entry I can’t see what it is.)

  24. On the last example, I corrected grammar on the paragraph ”The coequalizer of the pair of points…”, and (more importantly) I emphasized the key property that is necessary to construct and adjunction between standard spaces and superconvex spaces - mainly, for every superconvex space AA we need to contruct a universal arrow from the functor 𝒫\mathcal{P} to the object AA. My previous explanation was more confusing than helpful. Hence the last couple paragraphs of the last example have been deleted.

    Kirk Sturtz

    diff, v32, current

  25. This edit is for the purpose of answering, in part, Urs question - WHY DO WE CARE? Well we all know that the category of convex spaces is the category of algebras for the distribution monad on the category of Sets. Well that distribution monad can be viewed as arising from the adjunction given by the free space construction D:Set→CvxD:Set \rightarrow Cvx and the forgetful functor U:Cvx→SetU:Cvx \rightarrow Set. By replacing finite sums with countable affine sums we obtain the monad consisting of all countable affine sums on a set. The category of algebras for that monad is superconvex spaces.

    This only answers Urs question in part. To complete the answer I argue that if we want to understand (generalized) probability theory, then it is necessary to know the category of algebras for the Giry monad, say on the category of Standard Borel Spaces, which involves countability. Probability theory using the Kleisi category of the Giry monad does not shed any light on, for example, the probabilistic viewpoint of quantum mechanics. On the other hand, at least superconvex spaces allow us to understand probability amplitudes within the more generalized viewpoint using Algebras (=superconvex spaces).

    Kirk Sturtz

    diff, v33, current

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2024
    • (edited Jun 30th 2024)

    Thanks. Best to say this not at the end, but in the first lines of the Idea section.

    Something like: “The notion of superconvex spaces is a generalization of that of convex spaces which is useful for…”

    \,

    (In the entry I have made some cosmetic adjustments, such as of the vertical alignment of parenthesis and integral signs, in various formulas.)

    diff, v34, current

    • CommentRowNumber27.
    • CommentAuthorkirksturtz
    • CommentTimeJun 30th 2024
    Thanks for the adjustments Urs. I could add an additional line to the Idea section, and then leave the last example as is to explain the minor details. But I rather like the current Idea section that you wrote which stresses finite affine sums versus countable affine sums. I'll think about the possibility of adding another sentence to the Idea Section contrasting distributions with finite support to distributions with countable support. I agree it would shed more light on the WHY question.
    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeJun 30th 2024
    • (edited Jun 30th 2024)

    Just to say that I didn’t write the Idea section (just streamlined the wording a little, today): instead, its first line is yours from revision 1, and the second paragraph is due to “Anonymous” in revision 2 from July 2022.

  29. Per the last discussion on this page between myself and Urs, I put the result that the category of superconvex spaces is the category of algebras for the countable distribution monad in the Idea section, and delegated the result that it is the straightforward generalization of the result that convex spaces are the algebras for the finite distribution monad in the properties section. As a consequence, I removed the last example (which was previously this observation).

    In addition, I removed the first axiom (which follows from the second axiom) and also delegated that fact to the properties section.

    Kirk Sturtz

    diff, v36, current

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