Zinbiel 2-algebras

TL;DR

This paper introduces Zinbiel 2-algebras and 2-term Z-infinity algebras, proves their categorical equivalence, studies crossed module extensions, and explores their relationship with dendriform 2-algebras.

Contribution

It establishes the equivalence between Zinbiel 2-algebras and 2-term Z-infinity algebras and investigates their extensions and connections to dendriform 2-algebras.

Findings

Category of Zinbiel 2-algebras is equivalent to 2-term Z-infinity algebras.

Crossed module extensions of Zinbiel algebras are characterized.

Relationships between Zinbiel 2-algebras and dendriform 2-algebras are identified.

Abstract

The notions of Zinbiel 2-algebras and 2-term $Z_{\infty}$-algebras are introduced. It is proved that the category of Zinbiel 2-algebras and the category of $2$-term $Z_{\infty}$-algebras are equivalent to each other. Crossed module extensions of Zinbiel algebras are studied. Relationship between Zinbiel 2-algebras and dendriform 2-algebras are found.