Uniqueness of monoidal adjunctions

TL;DR

The paper addresses the problem of uniqueness for dual equivalences between the ∞-categories of ${\mathcal{O}}$-monoidal ∞-categories with right adjoint lax and left adjoint oplax monoidal functors. It introduces the space ${\mathcal{E}q}$ of equivalences compatible with monoidal presheaf functors ${\mathbb{P}}_{\mathcal{O}!}$ and ${\mathbb{P}}_{\mathcal{O}}^*$ and proves that ${\mathcal{E}q}$ is contractible, establishing a robust uniqueness result. It shows that the two known dual equivalences, one from Torii4 and the other from HHLN1, are canonically equivalent via a natural equivalence $(\gamma)$ making ${\mathbb{P}}_{\mathcal{O}}!\simeq {\mathbb{P}}_{\mathcal{O}}^*\circ H$. Consequently, the Torii4 equivalence $(T,\theta)$ is equivalent to the restriction $H$ of the HHLN1 equivalence. The work synthesizes higher-categorical adjunctions with presheaf formalism to yield a canonical identification between dual monoidal adjunctions, enhancing stability and transfer of results across models of ∞-categories using Day convolution and Yoneda-compatible data.

Abstract

There are two dual equivalences between the $\infty$-category of $\mathcal{O}$-monoidal $\infty$-categories with right adjoint lax $\mathcal{O}$-monoidal functors and that with left adjoint oplax $\mathcal{O}$-monoidal functors, where $\mathcal{O}$ is an $\infty$-operad. We study the space of equivalences between these two $\infty$-categories, and show that the two equivalences equipped with compatible $\mathcal{O}$-monoidal presheaf functors are canonically equivalent.

Theorems & Definitions (16)

• Theorem 1.1: cf. Theorem \ref{['thm:uniqueness-equivalences']}
• Theorem 1.2: cf. Theorem \ref{['thm:T-eq-H']}
• Corollary 1.3: cf. Corollary \ref{['cor:main-claim']}
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• Theorem 3.4
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3