As asked in the title but more specifically: does the nerve functor from Cat to sSet map a fibration between groupoids to a Kan fibration ? 


By fibration of groupoids I mean a fibration for the "natural" model structure where weak equivalences are categorical equivalences of groupoids, so by fibration I mean an isofibration. 


I don't think it's true in general but what about a fibration between fibrant groupoids ?


At the level of objects, does the nerve functor map fibrant groupoids to Kan complexes ?


Thanks



Does the nerve functor preserve fibrations?
Yes the nerve functor from groupoids to simplicial sets sends isofibrations to Kan fibrations. Being an isofibration means exactly that the nerve has the right lifting property against $\Lambda\_0^1\to\Delta^1$ and $\Lambda\_1^1\to \Delta^1$. The lifting property against higher dimensional horn inclusions is automatic.


You should also know that any groupoid is fibrant in the natural or canonical model structure.


