unital weak functor?

[John Baez <baez@math.ucr.edu>]

Josh Nichols-Barrer wrote:

> Hi everyone,
>
> Is there a name for a weak functor between bicategories which takes
> identity 1-morphisms to identity 1-morphisms?  "Unital weak functor"
> would
> seem an apt name, but if there is another with more precedent I'll just
> use that instead.

I don't think there's a standard term.

But, I like the term "normalized", since in certain circumstances weak
functors between
bicategories are described by cocycles in group cohomology, and the
cocycle is then
said to be "normalized" when the weak functor preserves identity
1-morphisms. It's an old
fact that every cocycle is equivalent to a normalized one, and this is
related to the fact
that every weak functor is isomorphic to a normalized one.

For more information on this, see:

Andre Joyal and Ross Street, Braided monoidal categories,
Macquarie Mathematics Report No. 860081, November 1986.
Also available at http://rutherglen.ics.mq.edu.au/~street/JS86.pdf

or for a pedagogical treatment, try section 8.3, "Classifying 2-groups
using
group cohomology", of this:

John Baez and Aaron Lauda, Higher-dimensional algebra V: 2-Groups,
Theory and Applications of Categories 12 (2004), 423-491.
Also available at http://arxiv.org/abs/math.QA/0307200

Best,
jb