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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.)
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