added a few references (starting here) by C. Okay et al on using the Giry monad extended to simplicial sets to characterize quantum contextuality via simplicial homotopy theory

]]>added publication data for this item:

- Tom Avery,
*Codensity and the Giry monad*, Journal of Pure and Applied Algebra**220**3 (2016) 1229-1251 [arXiv:1410.4432, doi:10.1016/j.jpaa.2015.08.017]

Creating link to new page for super convex spaces.

Kirk Sturtz

]]>I suggest that if you request an author hyperlink, as in

- Ruben Van Belle,
*Probability monads as codensity monads*(2021) $[$arXiv:2111.01250$]$

that you immediately go and create that author page (just a brief page: pointer to the author’s web presence plus the reference).

Because otherwise nobody will do, typically, in which case the broken link looks bad.

]]>I added the reference to Ruben Van Belle’s article which can be viewed, in hindsight, as naturally suggesting the two constructions (dense and codense functors) which arise when considering super convex spaces. The explanation in the main article (on $G$-algebras) is the easiest way to motivate the constructions and explain why the countable set $\mathbb{N}$ is sufficient”.

Kirk Sturtz

]]>In explaining why we are naturally led to super convex spaces I added the relatively short proof that every G-algebra specifies a super convex space and, likewise, that every map of G-algebras is a countably affine map.

Incidentally, I suspect that Doberkats category may have the Klesi category = category of Algebras because continuous map do not permit discrete spaces so everything is embeddable into a vector space which yields the algebras as $\mu$ = averages over all probability measures.

Kirk Sturtz

]]>Corrections/Modifications. In the section on Algebras over the Giry monad: (1) Doberkat’s example is the free algebra on $\mathbf{2} =\{0,1\}$. It is rather useless as an example precisely because it is free. This leads to the second correction, (2) There seems to be a fundamental misunderstanding why one necessarily needs a concrete representation. I hope the added paragraph explains the necessity (concerning existence of non free algebras). (3) I updated my reference to my latest work - which gives the simplest most direct way to show the existence of non free algebras using the support of a probability measure (=added Lemma) which I wish I understood 10 years ago!)

Kirk Sturtz

]]>I’ll guess what is meant is that you take the inverse images of measurable sets of $[0, 1]$, which gives you a $\sigma$-algebra on $G(X)$.

]]>In the formula for the multiplication of the Giry monad (why is it called the “counit”?), what is P(X) in the integral? P(X) is only defined well below this formula. It looks like the formula should actually say G(X), not P(X).

The relevant change is Revision 23 by K. Sturtz on October 30, 2018.

]]>Between #21 and #28, there was discussion of additions by Sturz. He’s now brought out a new paper on Giry algebras over measurable spaces. Any views?

- Kirk Sturz,
*The existence and utility of Giry algebras in probability theory*, (arXiv:2006.08290)

With all this attention being paid to categorical probability theory now, surely some expert out there could help.

]]>I corrected the monadic part of the history.

]]>I have taken the liberty of turning the footnote into an item in the list if references – because that’s what it is and how you are using it in the text. Then I replaced the unspecified “in a video comment” with actual pointer to that reference.

(Am relieved to hear that the Pentagon doesn’t disagree with the category of probabilistic mappings.)

]]>Added a transcript of Lawvere’s comments.

]]>I removed the reference to personal communication with me since we also have the video. Added that Lawvere’s appendix was intended as a framework for verification protocols on arms control

]]>Dmitri Pavlov wrote at #35:

I feel like some part is missing from this story. Why was it attached as an appendix to this unrelated report?

Apparently this appendix was intended to provide a reasonable framework for the development of verification protocols in the arms control context. We also have Lawvere on video telling the story himself! I will add this link to the page.

]]>have moved the “History”-section from being the second to being the second-to-last one (i.e. right before the references).

Changed the very first line of the entry from

The

Giry monad(Giry 80) …

to

]]>The

Giry monad(Giry 80, following Lawvere 62)

40: according to vague memory it was this Dortmund preprint Characterizing the Eilenberg-Moore algebras for a monad of stochastic relations, listed at Giry monad. In any case it was not from a journal but a preprint of similar look and the content fits.

]]>I remember discussing one Doberkat’s paper on stochastic stuff and Giry monads and generalizations with Roland Fridrich at the time, actually several years earlier. Later Doberkat had some new articles which touched the topic and I got hold of the book only later. Which was original viewed, I should look at time stamps in my old directories. I can not do in the very moment, but I will check this later, thanks.

]]>About Michèle Giry, I met her just once in Madame Ehresmann’s office in Amiens. I believe she became a secondary school teacher and then went into teacher training. She used to have a webpage in the University of Amiens but that link is now dead so I presume she has retired.

]]>This reference seems to have been added by Zoran in May 2013, which is over a year from the last Wayback machine capture, so it’s conceivable the items were renumbered, though *Coalgebraic Logic Over General Measurable Spaces – A Survey* is the best bet for what was meant. I may have a look tomorrow at the article to see if it is even relevant, if Zoran (or someone else) doesn’t pipe and correct me.

OK, so “item 5” in this Internet Archive capture is

- E.-E. Doberkat, Ch. Schubert: Coalgebraic Logic Over General Measurable Spaces – A Survey. Math. Struct. Comp. Sci. 21 (2), 2011, 175 - 234 (pdf). We discuss in this survey the generalization of stochastic Kripke models for general modal logics through predicate liftings for functors over general measurable spaces. Results on expressivity are derived, and it is shown that selection arguments permit incorporating the discussion of bisimilarity, provided the underlying spaces are assumed to be Polish.

which doesn’t look right. Looking at those entries that mention ’Eilenberg’ or ’Giry’, it’s hard to tell which, if any, our article is meant to be describing.

]]>Updating various references with journal links etc.

]]>It’s been there right from the start rev 1.

]]>