Weighted Tensor Products of Joyal Species, Graphs, and Charades

TL;DR

This paper introduces a new family of monoidal structures on Joyal species and charades, inspired by weighted algebraic products, and explores their categorical properties and applications.

Contribution

It develops weighted tensor products for Joyal species and charades, linking algebraic structures like derivational and Rota-Baxter algebras to monoidal graph categories.

Findings

New monoidal structures on Joyal species and charades

Categorical lifting of weighted derivations

Connections between algebraic and categorical frameworks

Abstract

Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota-Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graphs in a monoidal additive category. Weighted derivations are lifted to the categorical level.