From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6715 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Lawvere on probability distributions as a monad Date: Mon, 20 Jun 2011 15:52:50 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1308790898 5588 80.91.229.12 (23 Jun 2011 01:01:38 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 23 Jun 2011 01:01:38 +0000 (UTC) To: "categories" Original-X-From: majordomo@mlist.mta.ca Thu Jun 23 03:01:33 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.30]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QZYIU-0001ND-BZ for gsmc-categories@m.gmane.org; Thu, 23 Jun 2011 03:01:30 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43256) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1QZYGk-00010c-3w; Wed, 22 Jun 2011 21:59:42 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QZYGj-0007uX-4Q for categories-list@mlist.mta.ca; Wed, 22 Jun 2011 21:59:41 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6715 Archived-At: 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" To: Cc: Steve Vickers , Subject: Re: categories: Re: Lawvere on probability distributions as a mo= nad 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, =3D): 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, =3D): 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, =3D)) of the real-valued function g(-, c) on Y w/ resp. to the measure f(x, =3D= ) on B and call that real number (g.f)(x, c) . The slogans "Associativity =3D Fubini" and "Identity =3D 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/ ]