双范畴中的Yoneda引理

Yoneda引理作为范畴论的基本工具，其在双范畴框架中的表述引发了新的数学探讨1。在这一更抽象的结构中，双范畴的0-胞代表范畴，但无法直接访问其对象1。

为了在双范畴中推广Yoneda引理，研究者使用分布函子替代传统的预层1。Yoneda嵌入的classifying arrow对应于hom-profunctor的伴随1。该构造的关键性质包括：密性条件表现为左Kan扩张沿自身等于恒等映射1；在equipment框架中，伴随的单位需满足同构条件以保证Yoneda结构的良定义1。这些泛构造和2-胞的引入展示了Yoneda嵌入在双范畴中的密性、完全性和忠实性等结构性质1。

A recent exploration examines how the Yoneda lemma, a foundational tool in category theory, can be reformulated within the framework of double categories 1. The work addresses a key structural challenge: while 0-cells in double categories represent individual categories, they do not directly provide access to their objects 1. To overcome this limitation, the approach replaces the traditional use of presheaves—typically defined as Set-valued functors—with profunctors, requiring a reconceptualization of presheaves adapted to the double categorical setting 1.

The reformulation demonstrates how Yoneda embeddings function in this generalized context through the use of universal constructions and 2-cells 1. Central to the analysis is the observation that the classifying arrow of a Yoneda embedding corresponds to the adjoint of a hom-profunctor 1. A key property under examination is density: the left Kan extension of an embedding along itself must equal the identity map 1. Furthermore, when working within equipment structures, the unit of an adjoint must satisfy specific isomorphism conditions to ensure that the Yoneda structure remains well-defined 1.