TITLE: Nerve: Groupoids-> Kan Complexes.  Nerve: Bicategories w. adjoints -> ?
QUESTION [8 upvotes]: If you take the nerve of a groupoid, you get a Kan complex.
Question:
Take a bicategory that has adjoints for 1-morphisms, which is one notion of 'weak' groupoid (if all 2-morphisms are isomorphisms, then such a bicategory is a 2-groupoid), and take its nerve.
Is there a name for a bisimplicial set arising in this way?  Does it have some nice properties?  For example, is there a model structure on $\mathbf{ssSet}$ such that these are fibrant?

REPLY [2 votes]: I would recommend checking through the various papers by Cegarra and Remedios (look on the archive) They have done a lot of work in this area, but I am not sure if they have an answer for your question. The Duskin nerve as suggested by David Roberts is related to the bisimplicial approach and the relation is explored in various other papers from Granada, e.g. one by Manolo Bullejos and coauthors.