Fibrations of $\infty$-categories

Abstract

We construct a flagged $\infty$-category $\mathsf{Corr}$ of $\infty$-categories and bimodules among them. We prove that $\mathsf{Corr}$ 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 $\mathsf{LCorr}$ and $\mathsf{RCorr}$ of $\mathsf{Corr}$, 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.

89 pages, accepted by Higher Structures. Differs slightly from published version: Lemma 3.10 and Proposition 3.11 have been corrected

References in corpus (2)

• All about the Grothendieck construction
• A user's guide to co/cartesian fibrations

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