Coalgebras governing both weighted Hurwitz products and their pointwise transforms

TL;DR

This paper explores the algebraic structures underlying weighted Hurwitz and tensor products of Joyal species, revealing their connections to bialgebras, monoidal structures, and Dold–Kan type equivalences.

Contribution

It introduces a unified framework linking weighted Hurwitz products to bialgebras and extends these concepts to higher-dimensional linear algebra and monoidal categories.

Findings

Hurwitz product related to pointwise product and Rota--Baxter operators

Bialgebras obtained by freely generating from pointed coalgebras

Weighted Hurwitz monoidal structures on species category

Abstract

We give further insights into the weighted Hurwitz product and the weighted tensor product of Joyal species. Our first group of results relate the Hurwitz product to the pointwise product, including the interaction with Rota--Baxter operators. Our second group of results explain the first in terms of convolution with suitable bialgebras, and show that these bialgebras are in fact obtained in a particularly straightforward way by freely generating from pointed coalgebras. Our third group of results extend this from linear algebra to two-dimensional linear algebra, deriving the existence of weighted Hurwitz monoidal structures on the category of species using convolution with freely generated bimonoidales. Our final group of results relate Hurwitz monoidal structures with equivalences of of Dold--Kan type.