Institute of Mathematics

Abstract:  We construct a flagged $\infty$-category $\bf Corr$ of $\infty$-categories and bimodules among them. We prove that $\bf Corr$ 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 $\bf LCorr$ and $\bf RCorr$ of $\bf Corr$, whose morphisms are those bimodules which are {\it left-final} and {\it 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.