Subject: categories: re: co-

From: [email-masked] (Ross Street)
Date: Tue, 7 Jul 1998 10:49:43 +1000

While our insecurities about "co-" are being aired, I thought I should
admit to even more worries in the case of 2-categories (or bicategories)!
In these terminological matters, I have given up on linguistic correctness
and have also almost given up worrying about mathematical consistency.

Here is the difficulty. Motivation for 2-category theory comes from (at
least) two different directions which often lead to the same basic concepts
yet with different suggestions for terminology for the three other dual
concepts. Each concept has a co-, op-, and coop-version but the good choice
of op or co is not clear at all.

First motivation: We can take the view that our 2-category is foremost a
category with the 2-cells as extra structure (like homotopies in Top).
Then, for example, as pointed out by John Gray in the La Jolla 1965 volume,
Grothendieck was wrong in using "cofibration" for the *2-cell*-reversing
dual of fibration. Compare the situation in  Top  where cofibrations are
the *arrow*-reversing dual of fibrations.  So this leads to "opfibration"
for the *2-cell*-reversing dual of "fibration" (this is unnecessary in  Top
since homotopies are invertible).  However, Grothendieck's terminology has
stuck in some literature.  Using this first motivation, we define products
and coproducts of objects in a 2-category as we would in a category plus an
extra 2-cell condition.

Second motivation: We think of our 2-category  K  as a place to develop
category theory so that arrows f : U --> A  into an object  A  of  K  are
thought of as generalised objects of  A,  and 2-cells into  A  are
generalised arrows of  A.  Take a notion such as monad on  A.  From this
motivation, reversing *2-cells* in  K,  we should get the notion of
"comonad".  This terminology is in conflict with the doctrine developed on
the basis of the first motivation. Of course, "monad" is invariant under
*arrow*-reversal, but there are other concepts which are not.

--Ross