Dear Marta, I too am interested in seeing Bill's preprint, but is it indeed about the distributions in the sense of the symmetric topos? (Let us agree on calling those Lawvere distributions.) Although the formal analogies between Lawvere distributions and ordinary, measure theoretic, distributions are very profound, it seems to me it would be misleading to talk as if they were a categorical generalization of the ordinary ones. You and Funk have shown that, restricting to localic toposes, the localic reflection of the symmetric topos is the lower powerlocale. But that is not adequate in itself for describing probabilistic features. For that we need the valuation locale. I don't see that the symmetric topos has that extra measure-theoretic nature that is absent from the lower powerlocale and has to be added in the valuation locale. Roughly speaking, for the valuation locale, the lower powerlocale, and the symmetric topos, the extensive and intensive quantities take their values in the space R of, respectively, lower reals (0 to infinity inclusive), subsets of 1 (R = Sierpinski), and sets (R = object classifier). Your result about localic reflection then restricts to the fact that Sierpinski is the localic reflection of the object classifier. OK, I've indulged myself a bit with a spiel on something I've been thinking about, but there's a more straightforward question about the content of Bill's preprint. Is it (1) Lawvere distributions, (2) something more genuinely probabilistic, or (3) an account of how the former give us the latter? All the best, Steve. --- The valuation locale V(X) has as its points the continuous valuations on the frame of X, functions m from the frame to the lower reals from 0 to infinity, such that m is Scott continuous, m(0) = 0 and m has the modular law m(U)+m(V)=m(U\/V)+m(U/\V). It does not completely capture measure theory, but those aspects that can be derived from the measures on the opens of spaces. It relates to the probabilistic powerdomain of computer science. See - Vickers "A localic theory of lower and upper integrals" Coquand and Spitters "Integrals and valuations" Marta Bunge wrote:
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/ ************************************************
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