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

Josh Nichols-Barrer wrote:

...

Is there a name for a weak functor between bicategories which takes identity 1-morphisms to identity 1-morphisms?

The term 'normalised' (already mentioned in the replies) goes back at least to Grothendieck, SGA1, Exp VI: he calls a cleavage for a fibred category E -> B normalised if the cartesian lift of each identity arrow is an identity arrow -- this is exactly the condition for the correponding pseudo-functor B^op -> Cat to preserve identity arrows strictly. Joachim.