The binary product in the 2-category of triangular bialgebras and twisted morphisms

TL;DR

This paper extends the concept of binary products via tensor products from cocommutative bialgebras to the more general setting of triangular bialgebras and twisted morphisms, using 2-category frameworks.

Contribution

It introduces a new framework for understanding binary products in the context of triangular bialgebras and twisted morphisms, expanding the classical tensor product results.

Findings

Extended binary product concept to triangular bialgebras

Provided a new interpretation of twisted tensor products

Developed a framework using 2-categories and twists

Abstract

It is well known that the tensor product of two bialgebras constitutes the binary product in the category of cocommutative bialgebras and morphisms of bialgebras between them. In this paper, we extend this result to triangular bialgebras and twisted morphisms of triangular bialgebras. This is pursued by adopting the framework of 2-categories and the proper notion of binary product, as well as by employing a description of twists on the tensor product bialgebra, specifically developed for this purpose. We apply this extension to provide a new interpretation of the twisted tensor products of triangular bialgebras in terms of binary products.