(Weak) incidence bialgebras of monoidal categories

TL;DR

This paper explores how monoidal structures on categories induce (weak) bialgebra structures on their incidence coalgebras, connecting to combinatorial Hopf algebras and providing diverse examples.

Contribution

It introduces the concept of (weak) incidence bialgebras arising from monoidal categories and discusses their relation to existing algebraic frameworks and examples.

Findings

Incidence coalgebras can be endowed with (weak) bialgebra structures via monoidal products.

Connections are established between these structures and combinatorial Hopf algebras.

Examples include trees, skew shapes, bigraphs, and crossed modules.

Abstract

Incidence coalgebras of categories in the sense of Joni and Rota are studied, specifically cases where a monoidal product on the category turns these into (weak) bialgebras. The overlap with the theory of combinatorial Hopf algebras and that of Hopf quivers is discussed, and examples including trees, skew shapes, Milner's bigraphs and crossed modules are considered.