Cartesian Fibrations of $(\infty,2)$-categories

Abstract

In this article we introduce four variance flavours of cartesian 2-fibrations of $\infty$-bicategories with $\infty$-bicategorical fibres, in the framework of scaled simplicial sets. Given a map $p\colon \mathcal{E} \rightarrow\mathcal{B}$ of $\infty$-bicategories, we define $p$-(co)cartesian arrows and inner/outer triangles by means of lifting properties against $p$. Inner/outer (co)cartesian 2-fibrations are then defined to be 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 $\infty$-bicategories fibred in $\infty$-categories studied in previous work. We also prove that equivalences of such 2-fibrations can be tested fiberwise. As a motivating example, we show that the domain projection $\mathrm{d}\colon\mathrm{RMap}(\Delta^1,\mathcal{C})\rightarrow \mathcal{C}$ is a prototypical example of an outer cartesian 2-fibration, where $\mathrm{RMap}(X,Y)$ denotes the $\infty$-bicategory of functors, lax natural transformations and modifications. We then define inner/outer (co)cartesian 2-fibrations of categories enriched in $\infty$-categories, and we show that a fibration $p\colon \mathcal{E} \rightarrow \mathcal{B}$ of such categories is a (co)cartesian inner/outer 2-fibration if and only if the corresponding scaled nerve $\mathrm{N}^{\mathrm{sc}}(p)\colon \mathrm{N}^{\mathrm{sc}}\mathcal{E} \rightarrow \mathrm{N}^{\mathrm{sc}}\mathcal{B}$ is a fibration of this type between $\infty$-bicategories.