On the PROP corresponding to bialgebras

TL;DR

This paper constructs a PROP that characterizes bialgebras, providing a categorical framework to understand their algebraic structure through symmetric monoidal functors.

Contribution

It explicitly constructs a PROP whose algebras are exactly bialgebras, linking categorical structures with classical algebraic objects.

Findings

Explicit PROP construction for bialgebras

Equivalence between PROP algebras and bialgebras

Categorical framework for bialgebra theory

Abstract

A PROP is a symmetric monoidal category, whose set of objects is the set of natural numbers and on objects the monoidal structure is given by the addition. An algebra over a PROP is a symmetric strict monoidal functor to the tensor category of vector spaces. We give an explicite construction of the PROP whose category of algebras is equivalent to the category of bialgebras (= associative and coassociative bialgebras).