n-category = (n,n)-category
n-groupoid = (n,0)-category
We may think of the simplex category as the full subcategory of Cat on the categories free on non-empty finite linear graphs. This gives the canonical inclusion that defines the ordinary nerve of categories.
There is also the canonical embedding of categories into bicategories. Combined this gives the inclusion
The bicategorical nerve is the nerve induced from that. So far a bicategory we have
(Theorem 8.6 in (Duskin))
(This shows in particular that bigroupoids model all homotopy 2-types.)
Any strict 2-category determines both a ‘bicategory’ in the above sense (since a ‘strict’ thing is also a ‘weak’ one) and a simplicially enriched category. The latter is found by taking the nerve of each ‘hom-category’. The Duskin nerve of a 2-category is the same as the homotopy coherent nerve of the corresponding -category. This can also be applied to 2-groupoids and results in a classifying space construction for crossed modules.