Re: terminology

[selinger@mathstat.dal.ca (Peter Selinger)]

I had written:
> 
> My last comment is that, unlike what Jeff Egger claimed, "autonomous
> category" is not a special case of "*-autonomous category", because no
> symmetry is assumed in autonomous categories. Unless of course one
> first drops symmetry from the definition of *-autonomous categories,
> as Jeff has also suggested. As it stands, neither of "autonomous" and
> "*-autonomous" implies the other, which is perfectly fine in my
> opinion, since they are two different words.

I would like to clarify that Jeff himself did not say anything false,
because in the context in which he said it, he had in fact assumed the
non-symmetric definition of *-autonomous category (of [Barr 1995]).
Sorry if it sounded like I was accusing him.

My intention was only to point out that the statement "autonomous
categories are a special case of *-autonomous categories" cannot be
quoted out of context, because it is false under the original
definition of *-autonomous category that includes symmetry (of [Barr
1979]). Since it had already been quoted out of context when I wrote
the above, I just wanted to point out how the potential confusion. 

I think this is a very apt illustration of what happens if a term with
an existing meaning is redefined to mean something else. Henceforth it
is impossible for anybody to use the term (with either meaning)
without first giving a definition. That's no problem in a math paper,
where definitions are usually given or cited anyway, and therefore
terminology is in principle arbitrary. But it does tend to hobble
everyday discussion.

-- Peter

P.S.: since I have a demonstrated ability to put my foot in my mouth,
I'd like to clarify that I am not accusing Mike Barr of anything
either. His 1995 paper is clearly entitled "Non-symmetric *-autonomous
categories", and the inside of the paper clearly explains the
distinction. It is only in subsequent use that any confusion arises.
The usual solution, of putting either (non-symmetric) or (symmetric)
in parentheses the first time the term is used, and omitting it for
subsequent uses, is perfectly adequate. I am very happy with the
statement "an autonomous category is a special case of a
(non-symmetric) *-autonomous category".

M. Barr (1979). "*-Autonomous Categories", Lectures Notes in
Mathematics 752. Springer. 

M. Barr (1995). "Non-symmetric *-autonomous categories". 
Theoretical Computer Science 139:115–130.


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