Subject: categories: Re: co-

From: "Dr. P.T. Johnstone" <[email-masked]>
Date: Sat, 4 Jul 1998 18:30:40 +0100 (BST)

Of course the "co-" in "cofinal" is the Latin "cum", as it normally
is in English (if I refer to someone as my co-conspirator, I mean
he is conspiring with me, not against me!). But category-theorists
have got so firmly into the habit of using "co-" as an abbreviation
for "contra-" (except in the terms covariant and contravariant --
I assume they survived because they were widely used before categories
came along) that the "co-" in "cofinal" had to go. As for who killed
it off, the evidence points to Saunders Mac Lane as the guilty party
(see p. 213 of Categories for the Working Mathematician).

Category-theorists at least have the defence that the algebraic
topologists had started using "cohomology" for what should have been
"contrahomology" before categories came along. As Mike Barr mentioned,
Hilton and Wylie tried to encourage the use of "contrahomology" in
their book (1960), but it was probably far too late by then.

I'm surprised that no-one has yet mentioned Barry Mitchell's attempt,
in his book, to "eliminate the words left and right" from the language
of category theory. He did have a scheme for deciding which of a dual
pair of concepts should have the "co-"; unfortunately it led him to use
"adjoint" and "coadjoint" in the opposite sense to that in which most
people had been using them, and so much confusion resulted that everyone
went back to "left adjoint" and "right adjoint".

If it were possible to start afresh with the terminology of category
theory (of course it isn't, as Mike pointed out), I'd be in favour of
using "left" and "right" as much as possible, and eliminating the "co-"s.
(But even this is not guaranteed free from ambiguity. Has anyone apart
from me (and, I suppose, the authors) noticed that the usage of the
terms "left coset" and "right coset" in Mac Lane & Birkhoff's Algebra
is the opposite of that in Birkhoff & Mac Lane?)

Peter Johnstone

---

From: ZHAOD <[email-masked]>
Date: Mon, 13 Jul 1998 11:12:57 +0800

In mathematics many structures  are named 
after some one's name. For example, Boolean 
algebra, Hermitian matrices, Hausdorff 
space, Euclidean space. The first letters of 
the person's name are usually in upper case. 
The only exception I have seen are the 
abelian group and sober space. Does any one 
know why these two cases are different from 
others?

 Thanks

 Zhao Dongsheng

---

From: Michael Barr <[email-masked]>
Date: Mon, 13 Jul 1998 14:18:53 -0400 (EDT)

First off, sober (or sobre) is not an eponym (the fancy name for these).
I often write boolean, hausdorff, and euclidean.  You sometimes see
Abelian.  So there are no standards.  Note that boolean, euclidean and
abelian have adjectival endings (as does Hermitian), while hausdorff does
not.  In French and German, the two other languages I know somewhat, it is
standard that all adjectives be in lower case, that is not so in English.
So it comes down to a matter of convention and, basically, familiarity.
In physics, it is considered a mark of great respect to achieve lower case
status.  Nearly all physical constants and units are lower case.  But
mathematicians do not accept standards conventions and journals make no
attempt to enforce uniformity in these matters.