Cartesian fibrations of (∞,2)–categories

We introduce four variance flavors of (co)cartesian fibrations of 1-bicategories with 1-bicategorical fibers, in the framework of scaled simplicial sets.Given a map p W E !B of 1-bicategories, we define p-(co)cartesian arrows and inner/outer triangles by means of lifting properties against p, leading to a notion of 2-inner/outer (co)cartesian fibrations as those maps with enough (co)cartesian lifts for arrows and enough inner/outer lifts for triangles, together with a compatibility property with respect to whiskerings in the outer case.By doing so, we also recover in particular the case of 1-bicategories fibered in 1-categories studied in previous work.We also prove that equivalences of such fibrations can be tested fiberwise.As a motivating example, we show that the domain projection dW Fun gr . 1 ; C/ !C is a prototypical example of a 2-outer cartesian fibration, where Fun gr .X; Y / denotes the 1-bicategory of functors, lax natural transformations and modifications.We then define 2-inner and 2-outer flavors of (co)cartesian fibrations of categories enriched in 1-categories, and we show that a fibration p W E !B of such enriched categories is a (co)cartesian 2-inner/outer fibration if and only if the corresponding map N sc .p/WN sc E !N sc B is a fibration of this type between 1-bicategories.