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arXiv · 2508.11526

Revisiting $(\infty,2)$-naturality of the Yoneda embedding

Tobias Lenz
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Abstract

We show that the Yoneda embedding 'is' $(\infty,2)$-natural with respect to the functoriality of presheaves via left Kan extension, refining the $(\infty,1)$-categorical result proven independently by Haugseng-Hebestreit-Linskens-Nuiten and Ramzi, and answering a question of Ben-Moshe. As the key technical ingredient, we show that the identity functor of the $(\infty,1)$-category of $(\infty,1)$-categories admits only one enhancement to an $(\infty,2)$-functor (namely, the identity functor).

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Tobias Lenz. 2025-08-15. Revisiting $(\infty,2)$-naturality of the Yoneda embedding. https://doi.org/10.1112/blms.70254

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