Re: co-

["Dr. P.T. Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

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