2-Cartesian Fibrations I: A Model for $$\infty $$-Bicategories Fibred in $$\infty $$-Bicategories

Abstract

In this paper, we provide a notion of \(\infty \)-bicategories fibred in \(\infty \)-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled simplicial sets: Those are scaled simplicial sets equipped with an additional collection of triangles containing the scaled 2-simplices, which we call lean triangles, in addition to a collection of edges containing all degenerate 1-simplices. We prove the existence of a left proper combinatorial simplicial model category whose fibrant objects are precisely the 2-Cartesian fibrations over a chosen scaled simplicial set S. Over the terminal scaled simplicial set, this provides a new model structure modeling \(\infty \)-bicategories, which we show is Quillen equivalent to Lurie’s scaled simplicial set model. We conclude by providing a characterization of 2-Cartesian fibrations over an \(\infty \)-bicategory. This characterization then allows us to identify those 2-Cartesian fibrations arising as the coherent nerve of a fibration of \({\text {Set}}^+_{\Delta }\)-enriched categories, thus showing that our definition recovers the preexisting notions of fibred 2-categories.