# Nerve: Groupoids-> Kan Complexes. Nerve: Bicategories w. adjoints -> ?

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?

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Dear Alan, how do you define the nerve of a bicategory. – Harry Gindi Feb 16 '11 at 22:26
Well my deleted comment was silly. @Harry - he's taking the hom-wise nerve to get a (weakened) simplicial category and the the other nerve to get a bisimplicial set. Personally I would take the Duskin nerve, which is the '2-simplices are 2-commuting triangles, etc.' version. – David Roberts Feb 16 '11 at 22:34
@David: I thought you could only apply functors homwise in the case where the enrichment is strict (strict 2-categories are categories enriched in the cartesian monoidal category $Cat$.) – Harry Gindi Feb 16 '11 at 23:17
I just read about Duskin nerve on nLab. I'm not sure I want that because I would like to end up in bisimplicial sets. If there's a nice answer for Duskin nerve though, I would love to hear it! – Alan Wilder Feb 16 '11 at 23:37
What do you mean by "has adjoints for 1-morphisms"? Every morphism has a left adjoint? a right adjoint? both? either? or that composition with a given morphism induces an adjunction between hom-categories? – Steve Lack Feb 17 '11 at 4:49