Fibrations of ∞-categories

David Ayala, John Francis


We construct a flagged ∞-category Corr of ∞-categories and bimodules among them. We prove that 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 ∞-category of ∞-categories and its opposite, respectively. We introduce the flagged ∞-subcategories LCorr and RCorr of Corr, whose morphisms are those bimodules which are left-final and right-initial, respectively. We identify the notions of fibrations these flagged ∞-subcategories classify, and show that these ∞-categories carry universal left/right fibrations.


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