A counterexample in quasi-category theory

Abstract

We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0 0 -simplices, and a weak categorical equivalence, but which is not inner anodyne. This answers an open question of Joyal. Furthermore, we use this morphism to refute a plausible description of the class of fibrations in Joyal’s model structure for quasi-categories.