4 Apr 1998 16:07

Neil Ghani's question

```A week or two ago, Neil Ghani asked about natural transformations
between set-valued functors (I think they were set-valued, but anyway
that is what my answer refers to and is probably true for any reasonably
complete codomain category although a different argument would be
required), say a: F ---> G, such that for any arrow f: A ---> B of the
domain category, the square
aA
FA --------> GA
|            |
|            |
|Ff          |Gf
|            |
|            |
v     aB     v
FB --------> GB
is a pullback.  At the time, I sent Neil a private reply, but it
bounced for some reason.  (I said that that I thought that this
condition was reasonable only when restricted to monic f and then such
an a is called an elementary embedding.)  Then a couple of people
answered that it was called a cartesian arrow and I didn't try to resend
my answer.  Well, there is a simpler answer.  In that generality, such
an a is called a natural equivalence.  In other words, non-trivial
examples do not exist.

To see this, it is useful to translate it, using Yoneda, into the
following form.  As usual, I will say that of two classes E and M of
arrows in a category, E _|_ M (E is orthogonal to M) if in any diagram
e
A ----> B
|       |
```

5 Apr 1998 07:16

```
I wish to repeat my advice that the email address maths <at> su.oz.au will
soon vanish, since some of my colleagues are still using it. My current

maxk <at> maths.usyd.edu.au

Similarly Bob Walters has the new address

bob <at> maths.usyd.edu.au

Max Kelly.

```
5 Apr 1998 00:09

IPL Call For Participation (one week until hotel deadline)

```
[ NOTE: The last day for hotel reservations is APRIL 12. ]

CALL FOR PARTICIPATION

Workshop on Internet Programming Languages

in conjunction with the IEEE Computer Society
International Conference on Computer Languages 1998
Loyola University Chicago
Chicago, USA
http://www.math.luc.edu/iccl98

May 13, 1998

The Internet has long provided a global computing infrastructure but, for most
of its history, there has not been much interest in programming languages
tailored specifically to that infrastructure. More recently, the Web has
produced a widespread interest in global resources and, as a consequence, in
global programmability. It is now commonplace to discuss how programs can be
made to run effectively and securely over the Internet.

This workshop will provide a forum for the discussions of all aspects of
computer languages for wide-area systems, including specification languages,
programming languages, semantics, implementation technologies, and application
experience.

Organizing Committee
Henri Bal, Vrije Universiteit, The Netherlands, bal <at> cs.vu.nl
Boumediene Belkhouche, Tulane University, USA, bb <at> eecs.tulane.edu
```

5 Apr 1998 00:09

ICCL Call For Participation (one week until hotel deadline)

```
[ NOTE: The last day for hotel reservations is APRIL 12. ]

CALL FOR PARTICIPATION

IEEE Computer Society
1998 International Conference on Computer Languages

Loyola University
Chicago, USA, 14--16 May, 1998

the IEEE Computer Society Technical Committee on Computer Languages,
in cooperation with the ACM Special Interest Group on Programming Languages.

http://www.math.luc.edu/iccl98/

This is the sixth in a series of conferences devoted to all aspects of
computer languages, serving to bring together people broadly
interested in machine processable descriptions.  The hallmarks of ICCL
are diversity, openness to a wide range of linguistic research, and
international representation.  The focus is on new ideas in languages
and language technology which are innovative or experimental in
nature.

On May 13, a pre-conference Workshop on Internet Programming Languages
will be conducted at Loyola University.  Details are given at the end
of this announcement.

```

4 Apr 1998 17:42

a fantastic simplification of our subject

```I can't make Mike's proof type-check, so let me try the following:
For any set-valued functor, T,  from any category, let  a:O -> T
be the unique transformation from the empty functor. Isn't it
obviously cartesian? Has Mike shown all set-valued functors are
empty?

Good heavens.

```
4 Apr 1998 16:57

Re: Neil Ghani's question

```Mike Barr's answer to Neil Ghani's question is very pretty, but
unfortunately wrong. There are many examples of cartesian natural
transformations (e.g. between functors Set --> Set) which are neither
epic nor monic: for example, the natural transformation
(-) x A --> (-) x B induced by an arbitrary map A --> B.

The mistake in Mike's proof occurs when he says

> Let E be the equalizer of u and v
> and let h^B ---> E be any arrow.

The trouble is that E could be the zero functor, so that there may
not be any arrows h^B --> E.

Peter Johnstone

```
6 Apr 1998 11:43

Pure Mathematics Lectureships

```
Dear Colleagues,

I would be grateful if you could publicize the following jobs.

[Category Theory is represented here in Computer Science, and is also
used by the Pure Mathematicians, in a combined Maths and Comp Sci
department.]

Thanks,

Roy Crole.

-------------------------------------------------------------------

UNIVERSITY OF LEICESTER, UK

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE

Lectureships (Grade A/B) in Pure Mathematics (Two Posts)

Applications are invited for the above posts.  Applicants should have
a strong research record in any branch of Pure Mathematics.
Applications are particularly welcome from Mathematicians whose
research interests intersect with existing research strengths in the
Pure Mathematics group.  These are: Group Representation Theory,
Algebraic Topology, Representations of Algebras and Ring
Theory. Tenable from 1 September 1998.

Further particulars and application forms are available, by quoting
```

6 Apr 1998 04:06

Re: Neil Ghani's question

```I refer to Michael Barr's comments on Neil Ghani's question on cartesian
natural transformations. These have been much studied, especially in
computer science conrexts, and Michael admits he had seen the replies of
Peter Johnstone and myself, in which we independently give precise
references to our separate contributions to the study of monads whose
multiplication and unit are cartesian natural transformations; such as
the monad whose algebras are pointed sets, the multiplication for which
is the cartesian natural transformation whose A-component is A+1+1 --> A+1.

Accordingly I thought it odd that Michael, in the face of this, trusted
his proof that they exist only trivially. Of course, as Peter Johnstone
said, Michael's E is usually empty. Ironically, Michael is tha author of
a famous and striking paper on the point of the empty set, which inter
alia points out earlier errors of this kind on the part of others.

Max Kelly.

```
6 Apr 1998 05:15

Re: Naturality Squares and Pullbacks

```
>A natural transformation is an indexed family of arrows such that a
>certain diagram commutes. One could require a stronger condition,
>namely that the said diagram is a pullback. What would such a
>transformation be called? I'm sure I've seen this in the literature
>before but I cant remember where. Pointers?

Cartesion natural transformations data:F=>L into the list functor have
been used to represent the data-shape decomposition of many
data types of the form FX.

data_X
FX --------> LX
|  |         |
F! = 	|  |         | L! =
shape	|--          | length
|            |
F1 --------> L1
data_1 =
arity

Examples include tree types and array types. See, for example

<at> Article{Jay95b,
Author= cbj,
Title={A semantics for shape},
Journal={Science of Computer Programming},
Volume=25,
Year={1995},
Pages={251--283}
```

6 Apr 1998 10:22

AWOCA'98

```
CALL FOR PAPERS

------------------------------------------------------------------------

Ninth Australasian Workshop on Combinatorial Algorithms
(AWOCA'98)

July 27-30, 1998
Perth, Western Australia, Australia

------------------------------------------------------------------------

The Ninth Australasian Workshop on Combinatorial Algorithms (AWOCA'98)
will be held at the School of Computing of Curtin University of
Technology in Perth, Western Australia, Australia on July 27-30, 1998.
The workshop will follow the style of its predecessors: problem-oriented
papers, and an emphasis on informal discussion. To facilitate free
interchange of ideas, the number of participants is limited. If you wish
to attend, you should fill out the attached registration form and return
it together with your payment to:

AWOCA98
School of Computing
Curtin University of Technology
GPO Box 1987 U
Perth, WA, Australia
Phone: +61.9.9266 7647
```