Subject: categories: the origins and use of "co"

From: [email-masked] (Max Kelly)
Date: Thu, 10 Sep 1998 20:50:38 +1000 (EST)

Having been out of email contact since late June, what with travels and
hospitals, I have only now read the correspondence in early July on this
subject. I note that the only "co" in the Macquarie dictionary is that
coming from Latin "cum', of which the "co" in "cosine" is a special case,
in that it is short for "complement".

I don't recall ever knowing the origin of "colimit" and so on, but I can
report an interesting observation by Sammy Eilenberg (in private conversation,
when we were collaborating) on why projective limits are "limits" and 
inductive limits are "colimits", and not the reverse. His point was that
L is a limit in the category A iff, for each a in A, the set A(a,L) is a 
limit in Set; while C is a colimit in A iff, for each a in A, the set A(C,a)
is a limit in Set - not a colimit. So one needs limits in Set even to DEFINE
colimits in A. If you like, God made limits, while colimits are Menschenwerk.

As for "cofinal", where the "co" was originally of the "cum" type, meaning
that the subsequence was "equally final" with the whole sequence, and which
was "confinal" in German, we of the Sydney school - very likely at the same
time as others - saw it as hopelessly confusing in view of the categorical
use of "co", and deliberately changed it to "final" in our writings, with
"initial" for the dual. Note that the notion of a final functor works
beautifully even for WEIGHTED limits, as in my book on enriched categories.

Max Kelly.