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In this post, I was introduced to the monad of finitely supported measures.

$HX$ is the set of finitely supported measures on $X$, with monad structure defined as for the Giry monad.

I have three questions. This monad, $H$, is presented on Set in the post, but I am wondering if the category of groupoids supports this monad? What is the category of factorizations of this monad on Set and what is the category of factorizations of this monad on the category of groupoids? A factorization of a monad $M$ on category $C$ is a category $D$, and an adjunction $U,V$ between $C$ and $D$ that generates the monad $M$. What is the Eilenberg-Moore category for this monad on the category of groupoids?

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  • $\begingroup$ Algebras don't form the Kleisli category, but form the Eilenberg-Moore category. So the title of this question does not reflect the actual post. $\endgroup$
    – darij grinberg
    Commented Oct 8, 2018 at 17:33
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    $\begingroup$ That said, I find the question quite interesting. The algebras for the $H$ monad are something like affine spaces, except that instead of taking arbitrary affine combinations, you can only take convex combinations. So, some sort of abstraction of convexity. Has anyone seen it? $\endgroup$
    – darij grinberg
    Commented Oct 8, 2018 at 17:39
  • $\begingroup$ @darijgrinberg if you have a suggestion for the title, let me know. $\endgroup$
    – Ben Sprott
    Commented Oct 8, 2018 at 18:22
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    $\begingroup$ @darijgrinberg These spaces are repeatedly rediscovered under different names, as is the characterization of those that embed in vector spaces by a cancellation property. See the first part of this: arxiv.org/pdf/0903.5522.pdf $\endgroup$
    – Robert Furber
    Commented Oct 8, 2018 at 20:37
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    $\begingroup$ If anyone wants to write up an answer, I will award it. $\endgroup$
    – Ben Sprott
    Commented Oct 9, 2018 at 1:15

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It looks like Fritz addressed this a while ago in Convex Spaces I: Definition and Examples. It is $\mathcal{G}_{\mathrm{fin}}$-algebra shown under Definition 3.3.

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    $\begingroup$ I believe Fritz deals with probability measures, while algebras over the monad corresponding to all (finitely supported) measures are just semimodules over the semiring of nonnegative reals. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 8, 2021 at 19:54
  • $\begingroup$ I think the usual etiquitte is that, if someone else answers a question in a comment, and you write up your answer, then you make the answer CW so as not to get reputation points for someone else's work. There was a discussion on meta about this ages ago. $\endgroup$
    – David White
    Commented Apr 4 at 11:36

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