Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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.
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!
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”.
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.
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 $k$ 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 $\mathcal{G}X$, viewed as a superconvex space, is a free space. Hence $\mathcal{G}X/ \mathcal{R} \cong \mathcal{G}(X/\mathcal{S})$. 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
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.
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.)
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 $A$ we need to contruct a universal arrow from the functor $\mathcal{P}$ to the object $A$. My previous explanation was more confusing than helpful. Hence the last couple paragraphs of the last example have been deleted.
Kirk Sturtz
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 $D:Set \rightarrow Cvx$ and the forgetful functor $U:Cvx \rightarrow Set$. 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
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.)
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.
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
1 to 29 of 29