Pentactions and action representability in the category of reduced groups with action

TL;DR

This paper introduces the concept of pentactions in the category of reduced groups with action, showing how they can represent all actions on a perfect object with zero weak stabilizer.

Contribution

It defines pentactions and proves their role in action representability within the category of reduced groups with action, extending the understanding of actions in this algebraic context.

Findings

Pentactions form a structure in the category of reduced groups with action.

Perfect objects with zero weak stabilizer are action representable.

Pentactions can represent all actions on certain objects.

Abstract

A notion of pentaction of any object in the category $\mathbf{rGr}^{\bullet}$ of reduced groups with action is introduced. The operations are defined in the set $\mathsf{Pentact}(A)$ of pentactions of an object $A$ of $\mathbf{rGr}^{\bullet}$. It is proved that if an object $A$ is perfect with zero weak stabilizer in the sense defined in the paper, then $\mathsf{Pentact}(A)$ is an object of $\mathbf{rGr}^{\bullet}$, it has a derived action on $A$, the object $A$ is action representable and $\mathsf{Pentact}(A)$ represents all actions on $A$.