Free bifibrations of (∞,2)-categories, 2-simplicial objects and the walking adjunction

Abstract

In this work, we develop a fibrational approach to freely adjoining adjoints in an (∞,2)-category. We construct the universal bifibration obtained from a cocartesian fibration of (∞,2)-categories by adjoining cartesian lifts over a chosen class of 1-morphisms in the base. We then use this construction to provide an explicit model for freely adjoining adjoints, together with a zig-zag formula for the resulting mapping (∞,1)-categories. As applications, we give a model-independent proof of the universal property of the walking adjunction, providing an alternative proof of a theorem of Riehl--Verity, and establish the universal characterization of the simplex 2-category conjectured by Dyckerhoff--Kapranov--Schechtman--Soibelman.