Dear Jeremy Gibbons, Strangely enough, since I have worked "extensively" for years on Lawvere distributions, I cannot help you locate the preprint in question. But I can answer some of the questions you pose. For that, I will quote our book, Marta Bunge and Jonathon Funk, Singular Coverings of Toposes, LNM 1890, Springer 2006 and references therein. It can be downloaded from the web. There is indeed a (Kock-Zoberlein) monad M (with unit the "Dirac delta" on the 2-category Top_S of Grothendieck toposes (actually toposes E bounded over a base topos S), called the "symmetric monad", such that the S-points of M(E), for E a topos, is the category of distributions on E in the sense of Lawvere, that is, the category of cocontinuous functors E-->S and natural transformations between them. The existence of the symmetric topos was established by myself (Bunge 1995) using forcing topologies, and given an "algebraic" construction in (Bunge-Carboni, 1995), including the KZ-monad structure. Of course we study (Bunge-Fuk 2006) the M-algebras and several other matters. Concerning probability distributions, there is a classifying monad too. A probability distribution on a topos is a distribution that preserves the terminal object. It is shown (Bunge-Funk 2006, Proposition 8.2.6) that, for any S-topos E there is a subtopos T(E) of M(E) that classifies the probability distributions on E. This means that for any E there is an equivalence between the category of probability distributions on E and that geometric morphisms E--> T(E) over S. The Dirac delta E--> M(E) is an S-essential geometric morphism so given by a 3-tuple of S-adjoints delta_! adj delta^* adj delta_*. T(E) is "simply" the subtopos of M(E) given by the least topology forcing delta(1) --> 1 to be an isomorphism. There is also a construction of T(E) in terms of sites (Proposition 8.2.9, op.cit.) On the other hand, I know nothing about your comment
(and so probabilistic computations can be captured as a "computational effect").
so that, if such a preprint exists, I too would like to have it. I Will ask Lawvere when (or if) I see him in a month unless he replies himself to your query. With best wishes, Marta Bunge ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill UniversityBurnside Hall, Office 1005 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810/3800 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************
From: jeremy.gibbons@cs.ox.ac.uk Subject: categories: Re: Lawvere on probability distributions as a monad Date: Mon, 13 Jun 2011 22:28:31 +0100 To: categories@mta.ca
Dear all,
I sent the following request a few months ago, looking for a "preprint" of Lawvere's from 1962. I'm not the only one interested - I got five replies, off-list, of the form "If you find it, could I have a copy too, please?"
I thought I would try one last time. My paper that cites this has been accepted for publication, and I'm doing due diligence by trying my best to track down original sources!
Does anyone know where I can find a copy?
Thanks again, Jeremy
On 22 Feb 2011, at 14:08, Jeremy Gibbons wrote:
I wonder if you fine categorists could help me track down an old preprint?
Many people have written about probability distributions forming a monad (and so probabilistic computations can be captured as a "computational effect"). The reference trail goes back to
Michele Giry, "A Categorical Approach to Probability Theory", LNM 915:68-85, 1981
and thence to
F W Lawvere, "The Category of Probabilistic Mappings", preprint, 1962
I have Giry's paper, but can find no trace on the web of Lawvere's preprint. Does anyone know where I might find a copy? Might someone even have a copy that they would be prepared to scan?
Thanks, Jeremy
Jeremy.Gibbons@cs.ox.ac.uk Oxford University Department of Computer Science, Wolfson Building, Parks Road, Oxford OX1 3QD, UK. +44 1865 283521 http://www.cs.ox.ac.uk/people/jeremy.gibbons/
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