Subject: categories: Paper available

From: [email-masked] (Richard Blute)
Date: Fri, 27 Feb 1998 09:33:00 -0500

The following paper is available by anonymous ftp at triples.math.mcgill.ca
in the directory pub/blute as nuclear.ps.gz. It is also on Prakash Panangaden's
homepage at www-acaps.cs.mcgill.ca. Feel free to contact me if there
are any problems.

Cheers,
Rick Blute

            Nuclear and Trace Ideals in Tensored *-Categories
            ================================================= 

         Samson Abramsky                       Richard Blute
         Department of Computer Science        Department of Mathematics 
         University of Edinburgh                  and Statistics
         Edinburgh, Scotland                   University of Ottawa
                                               Ottawa, Ontario, Canada

                          Prakash Panangaden
                          Department of Computer Science
                          McGill University
                          Montreal, Quebec, Canada

        Presented to Mike Barr on the occasion of his 60th birthday.

                             Abstract                            
                             ========

We generalize the notion of nuclear maps from functional analysis by
defining nuclear ideals in tensored *-categories.  The motivation for
this study came from attempts to generalize the structure of the category
of relations to handle what might be called ``probabilistic relations''.
The compact closed structure associated with the category of relations 
does not generalize directly, instead one obtains nuclear ideals.

Most tensored *-categories have a large class of morphisms
which behave as if they were part of a compact closed category, i.e. they
allow one to transfer variables between the domain and the codomain.  We
introduce the notion of nuclear ideals to analyze these classes of
morphisms.  In compact closed categories, we see that all morphisms
are nuclear, and in the category of Hilbert spaces, the nuclear morphisms
are the Hilbert-Schmidt maps.

We also introduce two new examples of tensored *-categories, in which
integration plays the role of composition. In the first, morphisms are a 
special class of distributions, which we call tame distributions. 
We also introduce a category of probabilistic relations which was 
the original motivating example. 

Finally, we extend the recent work of Joyal, Street and Verity 
on traced monoidal categories to this setting by introducing the notion 
of a trace ideal. For a given symmetric monoidal category, it is not 
generally the case that arbitrary endomorphisms can be assigned a trace. 
However, we can find ideals in the category on which a trace can be 
defined satisfying equations analogous to those of Joyal, Street and 
Verity. We establish a close correspondence between nuclear ideals and 
trace ideals in a tensored *-category, suggested by the correspondence 
between Hilbert-Schmidt operators and trace operators on a Hilbert space. 
When we apply our notion of trace ideal to the category of Hilbert spaces, 
we obtain the usual trace of an endomorphism in the trace class.