Categories and weak equivalences of graded algebras

TL;DR

This paper explores the concept of weak equivalence in graded algebras, introducing a universal group for gradings and analyzing the categorical properties of related functors.

Contribution

It introduces the notion of weak equivalence of gradings, defines the universal grading group, and studies the categorical properties of support and universal group functors.

Findings

The universal grading group is unique for each grading.

Support functor is an oplax 2-functor.

Universal grading group functor has no adjoints.

Abstract

When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G, however it turns out that there is one distinguished group among them called the universal group of the grading. In this paper we study categories and functors related to the notion of weak equivalence of gradings. In particular, we introduce an oplax 2-functor that assigns to each grading its support and show that the universal grading group functor has neither left nor right adjoint.