Monoidal Kleisli Bicategories and the Arithmetic Product of Coloured Symmetric Sequences

TL;DR

This paper extends the arithmetic product to coloured symmetric sequences, establishing a normal oplax monoidal structure on their bicategory using monoidal double categories for coherence verification.

Contribution

It introduces a novel extension of the arithmetic product to coloured symmetric sequences and develops a framework using monoidal double categories for monoidal bicategory coherence.

Findings

Extended arithmetic product to coloured symmetric sequences

Established a normal oplax monoidal structure on the bicategory

Utilized monoidal double categories for coherence verification

Abstract

We extend the arithmetic product of species of structures and symmetric sequences studied by Maia and Mendez and by Dwyer and Hess to coloured symmetric sequences and show that it determines a normal oplax monoidal structure on the bicategory of coloured symmetric sequences. In order to do this, we establish general results on extending monoidal structures to Kleisli bicategories. Our approach uses monoidal double categories, which help us to attack the difficult problem of verifying the coherence conditions for a monoidal bicategory in an efficient way.