Duoidal categories, measuring comonoids and enrichment

TL;DR

This paper extends the theory of measuring comonoids to duoidal categories, establishing enriched structures and exploring their applications to graded objects, species, and operads.

Contribution

It introduces a framework for measuring comonoids within duoidal categories, enabling new enrichments of monoids, operads, and related structures.

Findings

Enrichment of monoids in comonoids via measuring comonoids

Construction of duoidal structures on categories of graded objects and species

Enrichment of operads in cooperads

Abstract

We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with an enrichment in the category of comonoids. The enriched homs are provided by the universal measuring comonoids. We study a number of duoidal structures on categories of graded objects and of species and the associated enriched categories, such as an enrichment of graded (twisted) monoids in graded (twisted) comonoids, as well as two enrichments of symmetric operads in symmetric cooperads.