Intrinsic Schreier special objects

TL;DR

This paper explores intrinsic Schreier split epimorphisms in a categorical setting, analyzing special objects with imaginary magma structures, their properties, and relations to protomodular objects and algebraic properties like the Engel property.

Contribution

It introduces and studies intrinsic Schreier special objects in a categorical framework, linking them to imaginary magma structures and algebraic properties.

Findings

Properties improve when imaginary magma structures are associative

Characterization of intrinsic Schreier special objects via imaginary magma structures

Relation to Engel property in groups and Lie algebras

Abstract

Motivated by the categorical-algebraic analysis of split epimorphisms of monoids, we study the concept of a special object induced by the intrinsic Schreier split epimorphisms in the context of a regular unital category with binary coproducts, comonadic covers and a natural imaginary splitting in the sense of our article [Intrinsic Schreier split extensions, Appl. Categ. Structures 28 (2020), 517--538]. In this context, each object comes naturally equipped with an imaginary magma structure. We analyse the intrinsic Schreier split epimorphisms in this setting, showing that their properties improve when the imaginary magma structures happen to be associative. We compare the intrinsic Schreier special objects with the protomodular objects, and characterise them in terms of the imaginary magma structure. We furthermore relate them to the Engel property in the case of groups and Lie algebras.