Para construction for double categories

TL;DR

This paper introduces Para and coPara double categories based on a horizontal action of a monoidal double category, providing characterizations of their monoidality properties and extending the structure of the original double category.

Contribution

It defines Para and coPara double categories for double categories and characterizes their monoidality in terms of the action and commutativity properties.

Findings

Characterization of monoidality of coPara_M(d)

Extension of monoidality from d to coPara_M(d)

Conditions for lax monoidality and premonoidality

Abstract

We introduce Para and coPara double categories for double categories. They rely on a horizontal action $\crta\ot$ of a horizontally monoidal double category $\Mm$ on a double category $\Dd$. We prove a series of properties, most importantly, we characterize monoidality of $\coPara_\Mm(\Dd)$ in the way that it extends monoidality of $\Dd$ as: lax monoidality of the action $\crta\ot$; bistrong commutativity of $\crta\ot$; and purely central premonoidality of $\coPara_\Mm(\Dd)$.