Subject: categories: Re: category theory and probability theory

From: F W Lawvere <[email-masked]>
Date: Sun, 25 Oct 1998 13:18:33 -0500 (EST)

In reply to the query of Jean-Pierre Cotton, I would like to mention the
following:
 
	In Springer LNM 915 (1982), an article by Michele Giry develops some
aspects of "A categorical approach to probability theory".The key idea,
which also was discussed in an unpublished 1962 paper of mine, is that
random maps between spaces are just maps in a category of convex spaces
between "simplices". There is a natural (semi) metric on the homs which
permits measuring the failure of diagrams to commute precisely, suggesting
statistical criteria. To make full use of the monoidal closed structure,as
well as to account for convex constraints on random maps,it seems
promising to consider also nonsimplices. (Noncategories is usually not a
good idea). The central observation that the metrizing process is actually
a monoidal functor was exploited in the unpublished doctoral thesis here
at Buffalo by X-Q Meng a few years ago in order to clarify statistical
decision procedures and stochastic processes as diagrams in a basic
convexity category. She can be reached at : [email-masked]   

Best wishes to those interested in pursuing this topic!
 
						Bill Lawvere

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F. William Lawvere			Mathematics Dept. SUNY 
[email-masked]               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

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On Tue, 20 Oct 1998, jean-pierre-C. wrote:

>   
>   Bonjour. I am a statistician and I should be interested in a categorical
> framework for probability and statistical theory. Does anyone know
> references (books, articles, websites...) about applications of categories
> and functors to probability or even measure theory ? Thank you.
>                                                
>                                            Very truly yours,  
>    
>                                            Jean-Pierre Cotton.
> 
> 
>