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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

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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.

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