Date: Tue, 20 Oct 1998 08:13:01 +0200 (DFT) From: "jean-pierre-C." Subject: categories: category theory and probability theory 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. From: "Amilcar Sernadas" Subject: Re: categories: category theory and probability theory Date: Wed, 21 Oct 1998 10:06:42 +0100 We are working on a related problem. It seems that it is necessary to work with a relaxed notion of category, namely where the compostion of f:a->b and g:b->c is not always defined. You should look at relaxed notions of category such as composition graphs, paracategories, precategories and the like. On our own preliminary results look at the working paper P. Mateus, A. Sernadas and C. Sernadas. Combining Probabilistic Automata: Categorial Characterization. Research Report, April 1998. Presented at the FIREworks Meeting, Magdeburg, May 15-16, 1998 that you can fetch from http://www.cs.math.ist.utl.pt/s84.www/cs/pmat.html Amilcar Sernadas -----Original Message----- From: jean-pierre-C. To: categories@mta.ca Date: Quarta-feira, 21 de Outubro de 1998 0:19 Subject: categories: category theory and probability theory > > 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. > Date: Sun, 25 Oct 1998 13:18:33 -0500 (EST) From: F W Lawvere Subject: categories: Re: category theory and probability theory 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develops some aspects of "A categorical approach to probability theory".The key idea, which also was discussed in an unpublished 1962 paper of mine, is that random maps between spaces are just maps in a category of convex spaces between "simplices".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 : meng@lmc.edu Best wishes to those interested in pursuing this topic! Bill Lawvere ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* 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. > > > From: boerger Date: Tue, 27 Oct 1998 15:52:52 +0100 Subject: categories: Re: category theory and probability theory I like to add some references concerning applications of category theory to measure theory: - Fred Lintonīs thesis and his paper "Functorial measure theory" in the Irvine Proceedings, Thompson , Washington D.C., !966. - Various papers by Mike Wendt, in particular his thesis on direct integrals of Hilbert spaces - My still unpublished long paper "Vector integration by universal properties" and its simplified version in the Bremen Proceedings, Heldermann, Berlin, 1991. Kind regards Reinhard