2-Chu

2-Chu

[Michael Barr]

I thought about what John Baez said.  No, I had nothing in mind for a
2-Chu, but now I do.  First, let me mention that Juergen Koslowski gave
a talk on 2-Chu at a meeting in Santa Barbara several years ago.

Anyway, what follows isn't a theory, only an example, not worked out in
detail.  Let us form what I will call Chu(Cat,Set) without getting
involved in size issues.  (It could, for example, be categories with
finite homsets and finite sets.)  An object would be (\C,\D,T) where \C
and \D are categories and T: \C x \D --> Set is a functor.  (I thought
of making it \C^o x \D, but in the made this choice.  It could be
changed.)  Anyway, a map in this category is (F,G,\alpha): (\C,\D,T) -->
(\C',\D',T') where F:\C --> \C', G: \D' --> \D are functors and \alpha:
T(-,G-) --> T'(F-,-) is a natural equivalence.

Make the class of maps (\C,\D,T) --> (\C',\D',T') into a category
[(\C,\D,T),(\C',\D',T')] with objects as above.  A morphism (F,G,\alpha)
--> (F',G',\alpha') is (\beta,\gamma), where \beta: F --> F' and \gamma:
G --> G' are natural transformations such that
                      \alpha(C,D')
            T(C,GD') -------------> T'(FC,D')
                |                        |
                |                        |
 T(C,\gamma D') |                        | T(\beta C,D')
                |                        |
                v                        v
            T(C,G'D') ------------> T'(F'C,D')
                      \alpha'(C,D')
 commutes. Now define
 (\C,\D,T) -o (\C',\D',T') = ([(\C,\D,T),(\C',\D',T')],\C x \D',S)
 with S either of the isomorphic pairings that takes
(F,G,\alpha)(C,D') to T(C,GD') or to T'(FC,D').  Of course, they are
isomorphic by \alpha(C,D'), but you still have to choose one or the
other.  Of course, there is also a choice involved in the direction of
\alpha.  The dual of an object interchanges the categories and replaces
\alpha by its inverse.  But you do need an isomorphism here, in order to
reverse things.