Actor of a crossed module of dialgebras via tetramultipliers

TL;DR

This paper investigates how actions in crossed modules of dialgebras can be represented using tetramultipliers, providing a construction for the actor and describing actions through equations.

Contribution

It introduces a method to construct the actor of a crossed module of dialgebras using tetramultipliers and characterizes actions explicitly.

Findings

Constructed the actor via a pair of dialgebras.

Provided a full description of actions through equations.

Showed the center equals the kernel of the canonical map under certain conditions.

Abstract

We study the representability of actions in the category of crossed modules of dialgebras via tetramultipliers. We deduce a pair of dialgebras in order to construct an object which, under certain circumstances, is the actor (also known as the split extension classifier). Moreover, we give give a full description of actions in terms of equations. Finally, we check that under the aforementioned circumstances, the center coincides with the kernel of the canonical map from a crossed module to its actor.