On the straightening of every functor

Abstract

We show that any functor between $\infty$-categories can be straightened. More precisely, we show that for any $\infty$-category $\mathcal{C}$, there is an equivalence between the $\infty$-category $({\mathrm{Cat}}_{\infty }{)}_{/\mathcal{C}}$ of $\infty$-categories over $\mathcal{C}$ and the $\infty$-category of unital lax functors from $\mathcal{C}$ to the double $\infty$-category $\mathrm{Corr}$ of correspondences. The proof relies on a certain universal property of the Morita category which is of independent interest.

arXiv commentv3: Made appendix into separate part of paper and added full details for the proof of Theorem B. v2: Added section 8 on comparison to other straightening equivalences, updated references, fixed typos. 28 pages. Comments welcome!