The uniqueness of braidings on the monoidal category of non-commutative descent data

TL;DR

This paper investigates the conditions under which the braiding on categories related to non-commutative descent data is unique, establishing that it is unique when a certain $k$-linear map exists, which holds in common algebraic cases.

Contribution

The paper proves the uniqueness of the braiding on categories of non-commutative descent data under specific algebraic conditions, extending understanding of their monoidal structures.

Findings

Braiding is unique if a $k$-linear unitary map $E : A o Z(A)$ exists.

Condition satisfied when $k$ is a field, or $A$ is commutative or separable.

Provides criteria for the uniqueness of braidings in non-commutative descent categories.

Abstract

Let $A$ be an algebra over a commutative ring $k$. It is known that the categories of non-commutative descent data, of comodules over the Sweedler canonical coring, of right $A$-modules with a flat connection are isomorphic as braided monoidal categories to the center of the category of $A$-bimodules. We prove that the braiding on these categories is unique if there exists a $k$-linear unitary map $E : A \to Z(A)$. This condition is satisfied if $k$ is a field or $A$ is a commutative or a separable algebra.