Erratum -- end of sentence after {\Integral} display in prior version -- corrected. Sorry. -- F. | To the Editor: if possible, suppress the earlier version and use this one instead, with these top 10 lines excised. Thanks, -- Fred ------ Original Message ------ Received: Mon, 20 Jun 2011 03:37:49 PM EDT From: "Fred E.J. Linton" <fejlinton@usa.net> To: <categories@mta.ca> Cc: Steve Vickers <s.j.vickers@cs.bham.ac.uk>, <martabunge@hotmail.com> Subject: Re: categories: Re: Lawvere on probability distributions as a monad As someone who for many years had a copy of that preprint, and may yet have a copy still, buried in the mess a rabid Dean made of the records in my office (by packing them all willy nilly in boxes and stacking those boxes into several six-foot tall columns in front of two bookshelves that are still holding books (!)), let me at least outline what I remember of it. [I might hope that one of the several folks I shipped copies of it to in years past might still have -- and share -- such a copy.] The work itself Bill developed in the early sixties, while both a grad student at Columbia and an employee of Litton Industries. The heart of it is a category -- rather Kleisli-category-like now, in retrospect, the way it's built, though the very notion of Kleisli category had not yet broken through the categorical consciousness -- whose objects, as I recall, were pairs made up of a set X and a boolean sigma-algebra A of subsets of X, while the maps from one such object (X, A) to another (Y, B) were those functions f: X --> prob(B) (from X to the set prob(B) of probability measures on B) for which, separately in each variable, each f(x, =): B --> R is a probability measure on B (yes, already said), each f(-, b): X --> R is an A-measurable real-valued function on X. For the composition of such an f with g: (Y, B) --> (Z, C), note that each f(x, =): B --> R is a probability measure on B and that each g(-, c): Y --> R is a B-measurable real-valued function on Y; so we may, for each x in X and c in C, form the integral (over B) {\Integral}_B g(-, c) d(f(x, =)) of the real-valued function g(-, c) on Y w/ resp. to the measure f(x, =) on B and call that real number (g.f)(x, c) . The slogans "Associativity = Fubini" and "Identity = Dirac Delta" outline how one sees this is a category. In the absence of the actual purple hexographed spirit document, I'm unable to reconstruct much more. But I hope this helps. And if Bill is tuning in to this thread, I'd be grateful if you could fine-tune what I've said, Bill, wherever I've gotten things off-pitch or out of key, and perhaps amplify what I've left too quiet. Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]