A double (∞,1)-categorical nerve for double categories
Abstract
We construct a nerve from double categories into double (∞,1)-categories and show that it gives a right Quillen and homotopically fully faithful functor between the model structure for weakly horizontally invariant double categories and the model structure on bisimplicial spaces for double (∞,1)-categories seen as double Segal objects in spaces complete in the horizontal direction. We then restrict the nerve along a homotopical horizontal embedding of 2-categories into double categories, and show that it gives a right Quillen and homotopically fully faithful functor between Lack's model structure for 2-categories and the model structure for 2-fold complete Segal spaces. We further show that Lack's model structure is right-induced along this nerve from the model structure for 2-fold complete Segal spaces.