Bicategories vs Rezk's weak 2-categories

Alexander Campbell – 30 May 2018, 6 June 2018

In this talk I will construct a Quillen equivalence between Lack's model category of bicategories and Rezk's model structure for weak 2-categories on the category of simplicial presheaves over Joyal's category Theta_2. The underlying adjunction of this Quillen equivalence is the composite of the homotopy coherent cellular nerve adjunction (defined in my talk of 24 January 2018) and an adjunction due to Ara.

To this end, I will construct the homotopy bicategory of a 2-quasi-category, which defines a left adjoint to the fully faithful homotopy coherent cellular nerve functor from the category of bicategories and normal pseudofunctors to the category of 2-quasi-categories, and show that a morphism of 2-quasi-categories is a weak equivalence in Ara's model structure iff it is essentially surjective on objects and an equivalence on hom-quasi-categories.