Fernando Abellán and Walker Stern
In this paper, we provide a notion of ∞-bicategories fibred in ∞-bicategories which we call 2-Cartesian fibrations. Our definition is formulated using the language of marked biscaled sim- plicial 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 ∞-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 ∞-bicategory.
