Adjunctions and Braided Objects

TL;DR

This paper explores the relationships between braided objects, algebras, and bialgebras in monoidal categories, introducing adjoint functors and constructions that deepen understanding of their algebraic structures.

Contribution

It constructs a braided primitive functor and a braided tensor bialgebra functor, establishing new adjunctions in the context of braided monoidal categories.

Findings

Defined a braided primitive functor.

Constructed a braided tensor bialgebra functor.

Analyzed functor behavior in braided categories.

Abstract

In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor and its left adjoint, the braided tensor bialgebra functor, from the category of braided objects to the one of braided bialgebras. The latter is obtained by a specific elaborated construction introducing a braided tensor algebra functor as a left adjoint of the forgetful functor from the category of braided algebras to the one of braided objects. The behaviour of these functors in the case when the base category is braided is also considered.