1 May 06:37 2001

Abelian Topological Groups


I am attempting to construct the ideal abelian category within which live
complete, hausdorff abelian topological groups.  The idea is that the
quotients of such a group, in the abelian category, would be completions
of the group with respect to topologies coarser than the given one.  The
subobjects would be those topologies.  Of course, having a topology as an
object in the abelian category means we have to have objects in the category
other than abelian groups.

Of course I want to know if this has been done before.  Also, what other ideas
are there about the ideal abelian category containing these groups?  Mac Lane
felt that compactly-generated spaces formed the ideal base category for
topological algebra.  I seem to be using the category of complete, hausdorff
uniform spaces as a base category.  I wrote a paper on (universal) algebras
with a compatible uniformity, and got some nice results about the congruence
(actually, uniformity) lattices.  But, admittedly, algebras with compatible
uniformities have drawbacks as a foundation for topological algebra because
even something like the complex numbers cannot be formalized as such, the
multiplication not being uniformly continuous.

Bill Rowan


2 May 12:11 2001

Re: Abelian Topological Groups

One thing is clear: your ideal abelian category is not abelian.
Furthermore, you don't get to choose your sub- and quotient objects; they
are imposed by the category.  Moreover, although a weaker topology (or an
abelian group with a weaker topology, which is what I assume is meant) is
certainly a subobject, it is not regular, which every subobject in an
abelian category must be.  In fact, the only abelian categories of
topological abelian groups I am aware of are the discrete groups and the
dual category of compact groups.  For me, the ideal category of
topological abelian groups is SP(LCA), the subobjects of products of
locally compact abelian groups.  It is not abelian, but it is
*-autonomous. It is equivalent to the category of weakly topologized
abelian groups or SP(R/Z), subobjects of powers of the circle.

On Mon, 30 Apr 2001, Bill Rowan wrote:

>
> I am attempting to construct the ideal abelian category within which live
> complete, hausdorff abelian topological groups.  The idea is that the
> quotients of such a group, in the abelian category, would be completions
> of the group with respect to topologies coarser than the given one.  The
> subobjects would be those topologies.  Of course, having a topology as an
> object in the abelian category means we have to have objects in the category
> other than abelian groups.
>
> Of course I want to know if this has been done before.  Also, what other ideas
> are there about the ideal abelian category containing these groups?  Mac Lane
> felt that compactly-generated spaces formed the ideal base category for
> topological algebra.  I seem to be using the category of complete, hausdorff
> uniform spaces as a base category.  I wrote a paper on (universal) algebras
> with a compatible uniformity, and got some nice results about the congruence


2 May 01:14 2001

Foundational Methods in Computer Science Workshop (FMCS'01)


ANNOUNCEMENT

Workshop on
Foundational Methods in Computer Science
--  a series of informal workshops on
categories and logic in computer science

FMCS'01 : May 31 -- June 3, 2001
Spokane, Washington

School of Electrical Engineering and Computer Science
Washington State University

May 31: Arrival day
-- Reception in the evening, 7-10pm PDT
June 1: Tutorial Talks
June 2: Research Talks
Conference Banquet
June 3: Research Talks in the morning
Noon -- end of workshop
[includes talks by John MacDonald, Robin Cockett, Ernie Manes, Paul Gilmore,
and tutorials by Ernie Manes and Phil Mulry]

WORKSHOP HOTEL:
WestCoast River Inn  (509) 326-5577
(800) 325-4000
State you are with the FMCS conference
to obtain the lower conference rate of \$US73 per night.


2 May 15:04 2001

Limits

Hi,
whenever I'm teaching basic category theory, students
ask me if there is a connection between limits in the
categorical sense and limits in the analytical sense,
e.g. the limit of a sequence of real numbers.
I've never found an answer to this question.

So I'd be very grateful for answers to one of the following:
- Can the limit of a sequence of real numbers be expressed
as a categorical limit (of course it can if the sequence is
monotone, but what if it is not)?
- Why have people chosen the term "limit" in category theory?
(And, by the way, who has defined it first?)

Tobias

--------------------------------------------------------------
Tobias Schröder
FB Mathematik und Informatik
Philipps-Universität Marburg
WWW: http://www.mathematik.uni-marburg.de/~tschroed
email: tschroed <at> mathematik.uni-marburg.de


1 May 18:47 2001

2001 Haskell Workshop: final call for papers

============================================================================

FINAL CALL FOR PAPERS

[Deadline for submission: 1st June 2001]

Firenze, Italy, 2nd September 2001

The Haskell Workshop forms part of the PLI 2001 colloquium
on Principles, Logics, and Implementations of high-level
programming languages, which comprises the ICFP/PPDP conferences
and associated workshops. Previous Haskell Workshops have been
held in La Jolla (1995), Amsterdam (1997), Paris (1999), and
Montreal (2000).

http://www.cs.uu.nl/people/ralf/hw2001.{html,pdf,ps,txt}

============================================================================

Scope
-----

The purpose of the Haskell Workshop is to discuss experience with
Haskell, and possible future developments for the language.  The scope
of the workshop includes all aspects of the design, semantics, theory,
application, implementation, and teaching of Haskell.  Submissions that
discuss limitations of Haskell at present and/or propose new ideas for


2 May 12:34 2001

diagxy.tex, final version

 % This is a package of commutative diagram macros built on top of Xy-pic
% by Michael Barr (email:  barr <at> barrs.org).  This may be freely
% distributed, unchanged, for non-commercial or commercial use.  If
% changed, it must be renamed.  Inclusion in a commercial software
package
% is also permitted, provided I receive one free copy of the software
% package for my personal use.  There are no guarantees that this package
% is good for anything.  I have tested it with LaTeX 2e, LaTeX 2.09 and
% Plain TeX.  Although I know of no reason it will not work with AMSTeX,
I
% have not tested it.

\input xy
\xyoption{arrow}

\newdir{ >}{{ }*!/-9pt/ <at> {>}}
\newdir{ (}{{ }*!/-5pt/ <at> {(}}
\newdir^{ (}{{ }*!/-5pt/ <at> ^{(}}
\newdir{< }{!/9pt/ <at> {<}*{ }}

\newbox\Label%
\newdimen\high%
\newdimen\deep%
\newdimen\ul%
\newcount\deltax%
\newcount\deltay%
\newcount\deltaX%
\newcount\deltaY%

\newdimen\wdth


2 May 12:37 2001

diaxydoc.tex

Note: This is not the version that will be posted, but rather for those
who have seen earlier versions.  But the examples are all the same.

\documentclass{article}
\input diagxy
\xyoption{curve}
\textwidth 6in
\oddsidemargin 0pt
\begin{document}
\def\xypic{\hbox{\rm\Xy-pic}}

\title{A new diagram package (Version 3)}
\author{Michael Barr\\Dept of Math and Stats\\McGill University\\805
Sherbrooke St. W\\Montreal, QC Canada H3A 2K6\\[5pt] barr <at> barrs.org}
\maketitle

\section*{Why a new diagram package?}

This started when a user of my old package, diagram, wrote to ask me if
dashed lines were possible.  The old package had dashed lines for
horizontal and vertical arrows, but not any other direction.  The reason
for this was that \LaTeX\ used rules for horizontal and vertical arrows,
but had its own fonts for other directions.  While rules could be made
any size, the smallest lines in other directions were too long for
decent looking dashes.  Presumably, Lamport was worried about compile
time and file size if the lines were too short, considerations that have
diminished over the years since.  Also arrows could be drawn in only 48
different directions, which is limiting.  My macros were not very well
implemented for slopes like 4 and 1/4, since I never used such lines.



2 May 19:10 2001

Re: Limits


Tobias Schroeder <tschroed <at> Mathematik.Uni-Marburg.de> writes:
> So I'd be very grateful for answers to one of the following:
> - Can the limit of a sequence of real numbers be expressed
>   as a categorical limit (of course it can if the sequence is
>   monotone, but what if it is not)?

With a little bit of cheating, you can use domain theory to express
the limit as a sequence as a _colimit_ in a partially ordered set.

Let D be the partial order consisting of all the closed intervals,
including singletons [a,a], ordered by reverse inclusion. We can
embed R into D by mapping it to the maximal elements a |---> [a,a],
and under a suitable topology on D (the Scott topology), this is
a topological embedding--purists may want to throw in R as the
smallest element to obtain an honest continuous domain.

Let x_i be a Cauchy sequence of real numbers. To say that x_i is a
Cauchy sequence is to say that there exist numbers d_i such that

(1) For j >= i, the interval [x_i - d_i, x_i + d_i]
contains [x_j + d_j, x_j + d_j].

(2) The numbers d_i become arbitrarily small: for every k
there is i such that for all j >= i, d_i < 1/k.

(Exercise for your students: show that this is equivalent to the usual
definition of Cauchy sequence.)

In terms of the partial order D, (1) says that the intervals


2 May 19:02 2001

Re: Limits

Tobias Schroeder asks:

- Can the limit of a sequence of real numbers be expressed
as a categorical limit (of course it can if the sequence is
monotone, but what if it is not)?

A good question. I have no answer, only a similar (and ancient)
question: is there a setting in which adjoint operators on Hilbert
spaces can be seen to be examples of adjoint functors between
categories?

As for his second question:

- Why have people chosen the term "limit" in category theory?
(And, by the way, who has defined it first?)

In the beginning, the only diagrams that had limits were "nets", that
is, diagrams based on directed posets. I believe it was Norman
Steenrod in his dissertation who first used the term. Before his
dissertation the Cech cohomology of a space was defined only as the
numberical invarients that arose as a limit of a directed set of such
invariants. It was Steenrod who perceived that Cech cohomology could
be defined as an abelian group. For that he needed to invent the
notion of a limit of a directed diagram of groups.

In the 50s the fact that one didn't need the diagram to be directed
was considered startling.

At least two of us tried to avoid the word "limit" in this more
general setting. Jim Lambek was pushing "inf" and "sup", a suggestion


3 May 14:59 2001

re: Limits

Tobias Schroeder writes:
> - Can the limit of a sequence of real numbers be expressed
>   as a categorical limit (of course it can if the sequence is
>   monotone, but what if it is not)?

I think I have an answer to this question (without cheating). It may
be well known or wrong (I haven't carefully checked the details, but I
believe that they are correct).

Given a metric space X with distance function d, construct a category,
also called X, as follows. The objects of the category X are the
points of the space X. An element of the hom-set X(x,y) is a triple
(r,x,y) with r a real number such that d(x,y)<=r. The composite of the
arrows r:x->y and s:y->z is the arrow s+r:x->z. This is well defined
by virtue of the triangle inequality d(x,z)<=d(x,y)+d(y,z). By virtue
of the condition d(x,x)=0, we have identities. Notice that all arrows
are mono.

Of course, because the category X is small and it is not a preordered
set, it doesn't have all limits. But some limits do exist.

CLAIM: Let x_n be a sequence of points of X, and, for each n, let the
arrow r_n:x_{n+1}->x_n be d(x_n,x_{n+1}). If the sum of r_k over k>=0
exists, then this omega^op-diagram has a categorical limit. The source
of the limiting cone is the metric limit l of the sequence. The
projection p_n:l->x_n is the sum of r_k over k>=n. If q_n:m->x_n is
another cone, then the mediating map u:l->m exists (and will be
automatically unique), because, by definition of cone and of our
category, q_n will have to be bigger than r_n, and then u=q_n-r_n does
the job.