Naturality of the $\infty$-Categorical Enriched Yoneda Embedding

TL;DR

This paper demonstrates the naturality of Hinich's $ abla$-categorical enriched Yoneda embedding by framing it as a unit of a partial adjunction, and explores atomic objects within tensored categories.

Contribution

It establishes the naturality of the enriched Yoneda embedding as a unit of a partial adjunction and investigates atomic objects in tensored categories.

Findings

The enriched Yoneda embedding is natural when viewed as a unit of a partial adjunction.

A finiteness condition called atomicity is introduced for objects in tensored categories.

The partial adjunction becomes a full adjunction when restricted to atomic objects.

Abstract

We make Hinich's $\infty$-categorical enriched Yoneda embedding natural. To do so, we exhibit it as the unit of a partial adjunction between the functor taking enriched presheaves and Heine's functor taking a tensored category to an enriched category. Furthermore, we study a finiteness condition of objects in a tensored category called being atomic, and show that the partial adjunction restricts to a (non-partial) adjunction between taking enriched presheaves and taking atomic objects.