Monoidal Adjunctions - Linearity and Duality

TL;DR

This paper explores how monoidal adjunctions between symmetric monoidal closed categories can be used to construct new such categories, extending existing work and revealing deep structural relationships.

Contribution

It introduces new constructions of symmetric monoidal closed categories from monoidal adjunctions, extending Kock's work and connecting to Chu-constructions.

Findings

Construction of symmetric monoidal closed structures on Eilenberg-Moore categories.

Development of categories al R_G and al L^F with monoidal closed structures.

Isomorphism between al L^F and al R_G under monoidal adjunctions.

Abstract

We explain two related constructions on the data of two monoidal symmetric closed categories $\mathscr{A}$ and $\mathscr{E}$ and monoidal functors $F: \mathscr{E}\to \mathscr{A}$ and $G: \mathscr{A}\to \mathscr{E}$. In a first part, we recall and partly extend work of A. Kock: In case $F$ is left-adjoint to $G$, and this adjunction is monoidal, we can equip the Eilenberg-Moore category $\mathscr{E}^T$ for $T$ being the canonical monad associated to the adjunction, with the structure of symmetric monoidal closed category, provided $\mathscr{E}$ has equalizers and $\mathscr{E}^T$ co-equalizers. In a second part, inspired by the Chu-construction, we build a category $\mathscr{R}_{G}$, which is symmetric monoidal closed as well, under the condition that $\mathscr{E}$ has pullbacks. Similarly we build a category $\mathscr{L}^{F}$ which is symmetric monoidal closed under the condition that $\mathscr{A}$ has what we call $F$-pushouts and $F$-pullbacks. In case $F \dashv G$ is a monoidal adjunction, we show that $\mathscr{L}^{F}$ and $\mathscr{R}_{G}$ are isomorphic as symmetric monoidal closed categories. We show also how $\mathscr{E}^T$ is related to both.