A Full and Faithful Nerve for 2-Categories

We prove that there is a full and faithful nerve functor defined on the category 2-Cat lax of 2-categories and (normal) lax 2-functors. This functor extends the usual notion of nerve of a category and it coincides on objects with the so-called geometric nerve of a 2-category or of a 2-groupoid. We also show that (normal) lax 2-natural transformations produce homotopies of a special kind, and that two lax 2-functors from a 2-category to a 2-groupoid have homotopic nerves if and only if there is a lax 2-natural transformation between them.