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.
Categories: math.CT math.AT