Dioperads, Frobenius monoidal functors and duality

TL;DR

This paper introduces the concept of a $d$-duality context in symmetric monoidal enriched categories, showing how Frobenius monoidal functors induce transformations between dioperad algebras, with implications for duality phenomena.

Contribution

It defines $d$-duality contexts, introduces $d$-twists of dioperads, and characterizes Frobenius monoidal functors as those inducing morphisms between dioperads.

Findings

Right adjoints in $d$-duality contexts map $ ext{dioperad}$-algebras to twisted algebras.

Frobenius monoidal functors induce morphisms between underlying dioperads.

Development of a dioperadic Day convolution and an $ ext{infinity}$-categorical extension.

Abstract

Motivated by duality phenomena for derived global sections on derived local systems on compact oriented manifolds, we introduce the notion of a $d$-duality context between symmetric monoidal enriched categories. In this setting, the right adjoint of a symmetric monoidal functor carries compatible lax and colax structures twisted by an invertible object $d$. For any enriched dioperad $\mathcal{P}$, we define a $d$-twist $\mathcal{P}\{d\}$ and prove that, in a $d$-duality context, the right adjoint sends $\mathcal{P}$-algebras to $\mathcal{P}\{-d\}$-algebras. To achieve this, the key conceptual result is that Frobenius monoidal functors between symmetric monoidal categories are precisely those functors inducing morphisms between the underlying dioperads. We also develop a dioperadic Day convolution, yielding an alternative proof of the main theorem and suggesting an $\infty$-categorical extension of the theory.