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Abstract Homotopy Theory Seminar
In this talk, I will explain a straightening equivalence for arbitrary functors between ∞-categories. Explicitly, this result states that for a given ∞-category C, there is a natural equivalence between the ∞-category of ∞-categories over C and the ∞-category of unital lax functors from C to a certain double ∞-category of correspondences. I will then describe some variations on this result: I will explain how to upgrade it to an equivalence of (∞,2)-categories and how one can deduce various classical straightening equivalences, such as those of Lurie and Ayala–Francis, from our result.