Paper available

Paper available

[Richard Blute]

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.

Re: Paper available

[Vaughan Pratt]

Vincent's paper

vs27@mcs.le.ac.uk wrote:
 > Hi Walter, let me advert my paper
 > on a similar and related subject
 > http://arxiv.org/abs/math/0602463
 > "Flatness, preorders and generalized metric spaces"
 > that treats completions of non symmetric spaces.
 > Cheers,
 > V.

reminds me of a question I've been meaning to ask for several years, in
fact since my CT'04 talk on communes over bimodules, but wasn't quite
sure how to formulate.

In any setting, ordinary or enriched, it is possible to introduce
presheaves immediately after defining "category," even before defining
"functor."

Ordinarily one does not do so because functors are more fundamental to
category theory than presheaves, being an essential stepping stone to
the notion of natural transformation, Mac Lane's motivating entity for
the whole CT enterprise.

But just as dessert tends to lose its appeal when complete demolition of
the main course is a prerequisite, so are applications of CT most
effective for a foreign (non-CT) audience when they don't assume that
the whole CT enchilada has been digested.  For applications of
presheaves it is helpful to know what is the absolute minimum of CT
required by the audience.

Just as it is not necessary to understand the principle of the internal
combustion engine when getting one's driver's license by showing that
one can control such an engine, it should not be necessary to know what
a functor, natural transformation, adjunction, or colimit is to freely
construct a presheaf on a small category J as a colimit.  The following
construction should suffice for those who know nothing more about CT
than the definition of category.

Grow a presheaf category C starting with C = J (with Set^{J^op} as the
unstated secret goal) as follows.  Independently adjoin objects x to C.
  For each such x and each object j in J, further adjoin morphisms from
j to x (more generally in the V-enriched case, assign an object of V to
C(j,x)), with composites of the morphisms of C(j,x) with those of J
chosen subject only to the requirement that C remain a category.  For
any objects x,y of C, with x not in J (y in J is ok), populate C(x,y)
with as many morphisms f,g,... as possible (in the V-enriched case, a
suitable limit), again choosing composites with morphisms from any j to
x arbitrarily, subject to the requirements that (i) if for all j and all
morphisms a: j --> x, fa = ga, then f = g, and (ii) again that C remain
a category (which then determines all remaining composites x --> y -->
z).  A pre-question here is, did I inadvertently leave anything out?

My main question is, is there a reference for this process that I can
cite?  Any such reference must make the point that the prerequisites for
this process include categories but exclude the rest of CT (as
prerequisites---obviously some additional parts of CT are directly
derivable, the point is that they're not prerequisites for the student).

Ordinarily one reason for not bothering with such a thing would be that
one can avoid even the categories by talking about equational theories
with only unary operations.  My application however is to communes,
which are trickier to describe from a purely algebraic perspective
(they're chupological rather than coalgebraic), but very natural from
the above colimit-based perspective.

Vaughan

Re: Paper available

[vs27]

Hi Walter, let me advert my paper
on a similar and related subject
http://arxiv.org/abs/math/0602463
"Flatness, preorders and generalized metric spaces"
that treats completions of non symmetric spaces.
It took some time to be written but it is going to
appear (in an improved version) in the
Georgian Mathematical Journal.

Cheers,
V.


On Jan 9 2009, Walter Tholen wrote:

>The paper
>"On the categorical meaning of Hausdorff and Gromov distances, I"
>by Andrei Akhvlediani, Maria Manuel Clementino and myself is available at
>http://arxiv.org/abs/0901.0618
>and on my homepage at
>http://math.yorku.ca/~tholen/
>
>The paper expands on ideas offered in Maria Manuel Clementino's talk at
>CT08 and in my talks at the Octoberfest in Montreal and at the
>Borceux-Bourn Birthday Conference in Brussels in October.
>We welcome comments.
>
>Regards,
>Walter Tholen.
>
>
>
>
>
>

Paper available

[Walter Tholen]

The paper
"On the categorical meaning of Hausdorff and Gromov distances, I"
by Andrei Akhvlediani, Maria Manuel Clementino and myself is available at
http://arxiv.org/abs/0901.0618
and on my homepage at
http://math.yorku.ca/~tholen/

The paper expands on ideas offered in Maria Manuel Clementino's talk at
CT08 and in my talks at the Octoberfest in Montreal and at the
Borceux-Bourn Birthday Conference in Brussels in October.
We welcome comments.

Regards,
Walter Tholen.

Paper available

[Michael A. Warren]

Dear categorists,

The following paper, in which perhaps some of you might have an
interest, is now available:

"Homotopy theoretic models of identity types" by Steve Awodey and
Michael A. Warren

Abstract:

"This paper presents a novel connection between homotopical algebra
and mathematical logic.  It is shown that a form of intensional type
theory is valid in any Quillen model category, generalizing the
Hofmann-Streicher groupoid model of Martin-Loef type theory."

The paper may be found on the arXiv as 0709.0248v1 (math.LO):

<http://arxiv.org/abs/0709.0248>


Best regards,

Michael Warren

Paper available

[Walter Tholen]

The paper

"Torsion theories and radicals in normal categories"
by M.M. Clementino, D. Dikranjan, and W. Tholen

is available (pdf file) at

http://www.math.yorku.ca/Who/Faculty/Tholen/research.html

Comments welcome!

Walter Tholen.

Paper available

[Marta Bunge]

This is to announce a new paper, by

   Marta Bunge and Marcelo Fiore, "Unique factorization Lifting Functors
and Categories of Processes".

        http://www.dcs.ed.ac.uk/~mf/CONCURRENCY/ufl.dvi

	http://www.dcs.ed.ac.uk/~mf/CONCURRENCY/ufl.ps


The paper is organised as follows. After an Introduction, Section 1
presents background material motivated from the point of view of computer
science.  In Section 2, the category UFL of unique factorisation lifting
(ufl) functors is recalled and its basic properties are studied. Section 3
explores applications of ufl functors to concurrency.  In particular we
show that they may be used in the study of interleaving models like
transition systems.  In Section 4, we introduce triangulated categories.
Our main use for them is in Section 5 where, for C a triangulated category,
we exhibit the category UFL/C as a sheaf topos. These toposes may be
regarded as models of linearly-controlled processes.  Some
concluding remarks are provided in Section 6.









Professor Marta Bunge
McGill University
Department of Mathematics & Statistics
Burnside Hall
805 Sherbrooke Street West
Montreal, QC
Canada H3A 2K6

Fax: (514) 933 8741
Phone: (514) 933 6191