Fibrations of $\infty$-categories

Abstract

We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by Lurie of both coCartesian fibrations and Cartesian fibrations, as they are classified by the $\infty$-category of $\infty$-categories and its opposite, respectively. We introduce the flagged $\infty$-subcategories ${\sf LCorr}$ and ${\sf RCorr}$ of ${\sf Corr}$, whose morphisms are those bimodules which are \emph{left final} and \emph{right initial}, respectively. We identify the notions of fibrations these flagged $\infty$-subcategories classify, and show that these $\infty$-categories carry universal left/right fibrations.