On the straightening of every functor
Thomas Blom
TL;DR
This work proves that every functor between ∞-categories can be straightened: for any ∞-category $\mathcal{C}$ there is a natural equivalence $(\mathrm{Cat}_{\infty})_{/\mathcal{C}} \simeq \mathrm{Lax}_{\mathrm{un}}(\mathcal{C}^{\mathrm{h}}, \mathbb{C}\mathrm{orr})$, where $\mathbb{C}\mathrm{orr}$ is a double ∞-category of correspondences and lax functors are unital. The proof builds a universal Morita double category, showing that $\mathrm{Lax}(\mathbb{E}, \mathbb{D}) \simeq \mathrm{Lax}_{\mathrm{un}}(\mathbb{E}, \mathrm{Mor}(\mathbb{D}))$ for Moritable double categories $\mathbb{D}$, and then applies this to the span double category to obtain the main straightening result. By identifying $\mathbb{C}\mathrm{orr}$ with various other notions of correspondences (Ayala–Francis, Ruit, Heine) the paper unifies several strands of straightening results, including Conduché fibrations and locally cocartesian fibrations, within a single double‑categorical framework. The approach yields a clean, intrinsic equivalence that clarifies the role of correspondences in straightening and opens avenues for higher-categorical enhancements and extensions to $(\infty,n)$-categories. Overall, the work provides a robust, conceptual bridge between overcategories of ∞-categories and lax functors into a universal correspondence double category, with broad implications for higher Morita theory and fibration straightening.
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.