Colimits in enriched $\infty$-categories and Day convolution

TL;DR

This paper investigates colimits in enriched $al$-categories, establishing universal properties of the enriched Yoneda embedding and conditions under which enriched presheaf categories inherit monoidal structures.

Contribution

It proves the universality of the enriched Yoneda embedding and shows when enriched presheaf categories inherit monoidal structures in the context of $al$-categories.

Findings

Enriched Yoneda embedding is universal for $al$-functors.

Enriched presheaf categories inherit monoidal structures under certain conditions.

The study extends colimit theory to enriched $al$-categories.

Abstract

For a monoidal $\infty$-category $\mathcal{M}$ with colimits, we study colimits of $\mathcal{M}$-functors $\mathcal{A}\to\mathcal{B}$ where $\mathcal{B}$ is left-tensored over $\mathcal{M}$ and $\mathcal{A}$ is an $\mathcal{M}$-enriched category. We prove that the enriched Yoneda embedding $Y:\mathcal{A}\to P_{\mathcal{M}}(\mathcal{A})$ yields a universal $\mathcal{M}$-functor and, in the case when $\mathcal{A}$ has a certain monoidal structure, the category of enriched presheaves $P_{\mathcal{M}}(\mathcal{A})$ inherits the same monoidal structure.