Enhanced twisted arrow categories

Abstract

Given an $\infty$-bicategory $\mathbb{D}$ with underlying $\infty$-category $\mathcal{D}$, we construct a Cartesian fibration $\mathrm{Tw}(\mathbb{D})\to \mathcal{D}\times {\mathcal{D}}^{\mathrm{op}}$, which we call the enhanced twisted arrow $\infty$-category, classifying the restricted mapping category functor ${\mathrm{Map}}_{\mathbb{D}}:{\mathcal{D}}^{\mathrm{op}}\times \mathcal{D}\to {\mathbb{D}}^{\mathrm{op}}\times \mathbb{D}\to {\mathrm{Cat}}_{\infty }$. With the aid of this new construction, we provide a description of the $\infty$-category of natural transformations $\mathrm{Nat}(F,G)$ as an end for any functors $F$ and $G$ from an $\infty$-category to an $\infty$-bicategory. As an application of our results, we demonstrate that the definition of weighted colimits presented in arXiv:1501.02161 satisfies the expected 2-dimensional universal property.

35 pages

References in corpus (2)

• Higher Topos Theory
• Dualizing cartesian and cocartesian fibrations