Re: Lawvere on probability distributions as a monad

[Marta Bunge <martabunge@hotmail.com>]

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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