Enhanced twisted arrow categories

TL;DR

This paper introduces an enhanced twisted arrow $b$-category for $b$-categories, providing a new way to describe natural transformations and verifying a key universal property of weighted colimits.

Contribution

It constructs the enhanced twisted arrow $b$-category and uses it to describe natural transformations as ends, advancing the understanding of $b$-categorical limits and colimits.

Findings

Provides a new Cartesian fibration for $b$-categories

Describes natural transformations as ends using the new construction

Confirms the universal property of weighted colimits

Abstract

Given an $\infty$-bicategory $\mathbb{D}$ with underlying $\infty$-category $\mathcal{D}$, we construct a Cartesian fibration $\operatorname{Tw}(\mathbb{D})\to \mathcal{D} \times \mathcal{D}^{\operatorname{op}}$, which we call the enhanced twisted arrow $\infty$-category, classifying the restricted mapping category functor $\operatorname{Map}_{\mathbb{D}}:\mathcal{D}^{\operatorname{op}}\times \mathcal{D} \to \mathbb{D}^{\operatorname{op}} \times \mathbb{D} \to \operatorname{Cat}_{\infty}$. With the aid of this new construction, we provide a description of the $\infty$-category of natural transformations $\operatorname{Nat}(F,G)$ 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.