Pivotal Objects in Monoidal Categories and Their Hopf Monads

TL;DR

This paper introduces pivotal objects in monoidal categories, constructs associated categories with monadic and Hopf monad structures, and extends the framework to pivotal diagrams, enriching the theory of monoidal categories.

Contribution

It constructs new categories from pivotal objects, demonstrates their monadic and Hopf monad structures, and extends the theory to pivotal diagrams in monoidal categories.

Findings

The category $ ext{C}(P,Q)$ lifts the monoidal and closed structures of $ ext{C}$.

Under suitable colimits, $ ext{C}(P,Q)$ is monadic, leading to Hopf monads.

Extended the framework to arbitrary pivotal diagrams in monoidal categories.

Abstract

An object $P$ in a monoidal category $\mathcal{C}$ is called pivotal if its left dual and right dual objects are isomorphic. Given such an object and a choice of dual $Q$, we construct the category $\mathcal{C}(P,Q)$, of objects which intertwine with $P$ and $Q$ in a compatible manner. We show that this category lifts the monoidal structure of $\mathcal{C}$ and the closed structure of $\mathcal{C}$, when $\mathcal{C}$ is closed. If $\mathcal{C}$ has suitable colimits we show that $\mathcal{C}(P,Q)$ is monadic and thereby construct a family of Hopf monads on arbitrary closed monoidal categories $\mathcal{C}$. We also introduce the pivotal cover of a monoidal category and extend our work to arbitrary pivotal diagrams.