categories: re: adjoint equivalence

[baez@math.ucr.edu]

Paul Levy writes:

> Hi, I have a question.  If P and Q are objects in a 2-category C, and 
> there is an equivalence between them, must there be an adjoint 
> equivalence (an adjunction whose unit and counit are both isomorphisms) 
> between them?  

Yes, and constructing this adjoint equivalence is an incredibly 
fun exercise in playing around with diagrams for 2-morphisms in 
your 2-category!  The proof must appear in the literature, but 
I don't know where, and it's really much better to do this sort
of thing oneself.  Knowing that it's possible should give you the
gumption to do it.  But if you get stuck, you can find the basic 
trick in the proof of Prop. 27 in my paper "Higher-dimensional 
algebra II: 2-Hilbert Spaces", which is available at

http://xxx.lanl.gov/abs/q-alg/9609018

Ignore the rather complicated context and just stare at the formulas.

Best,
jb