An equivalence between two models of $\infty$-categories of enriched presheaves

TL;DR

This paper establishes an equivalence between two models of $ abla$-categories of enriched presheaves, demonstrating their conceptual and practical interchangeability in the context of $ abla$-enriched $ abla$-categories.

Contribution

The authors construct an explicit equivalence between Hinich's and their own models of $ abla$-categories of enriched presheaves, clarifying their relationship.

Findings

Proves an equivalence of $ abla$-categories of enriched functors

Shows models are weakly right tensored over $ abla$-operads

Bridges two different approaches to $ abla$-categories

Abstract

Let $\mathcal{O} \to \mathrm{BM}$ be a $ \mathrm{BM}$-operad that exhibits an $\infty$-category $\mathcal{D}$ as weakly bitensored over non-symmetric $\infty$-operads $\mathcal{V}, \mathcal{W}$ and $\mathcal{C}$ a $\mathcal{V}$-enriched $\infty$-precategory. We construct an equivalence $$\mathrm{Fun}_{\mathrm{Hin}}^{\mathcal{V}}(\mathcal{C},\mathcal{D}) \simeq \mathrm{Fun}^{\mathcal{V}}(\mathcal{C},\mathcal{D}) $$ of $\infty$-categories weakly right tensored over $\mathcal{W}$ between two different models of $\infty$-categories of $\mathcal{V}$-enriched functors, one introduced by Hinich and one constructed by us.