Subject: categories: re: Equivalences and psuedo-equivalences (two items)

From: Steve Lack <[email-masked]>
Date: Wed, 25 Feb 2004 09:34:18 +1100

Carl Futia asks the following questions:

(1) If F:A-->B is a biequivalence of bicategories, is it
equivalent to a strict homomorphism?
(2) As in (1), but suppose that A and B are 2-categories.

The answer to both questions is no. Here's an example. Let
A be the 2-element group {0,1}, seen as a 2-category with one
object, two arrows, and no non-trivial 2-cells. Let B be the
2-category with one object, with the integers as arrows (and
composition given by addition) and with a unique, invertible
2-cell between arrows m and n if m-n is even, and no other
2-cells. The only strict homormorphism (i.e. 2-functor) from
A to B sends both arrows of A to the identity arrow 0 of B.
There is, however, a biequivalence F:A-->B, sending 0 to 0
and 1 to 1.

Note also that there is an obvious 2-functor G:B-->A which is
a biequivalence. So the example also illustrates that for a
2-functor which is a biequivalence it may not be possible to
choose an ``inverse biequivalence'' which is a 2-functor.

This example appeared in:

     Stephen Lack, A Quillen model structure for 2-categories,
     K-Theory 26:171-205, 2002

as Example 3.1 on page 178. In that context G:B-->A is
actually a trivial fibration, so the fact that there is
no 2-functor F with GF=1 also shows that the 2-category A
is not cofibrant.

Those interested in the rest of the paper should also look
at the sequel:

   Stephen Lack, A Quillen model structure for bicategories,
   available from
   http://www.maths.usyd.edu.au/u/stevel/papers/qmcbicat.html

which corrects an error in the model structure definition
given in the earlier paper, and also extends the model
structure to bicategories, giving a Quillen equivalence
between the two model categories.

Steve Lack.