Higher categorified algebras versus bounded homotopy algebras

TL;DR

This paper introduces Lie 3-algebras, establishes their correspondence with specific 3-term Lie infinity algebras, and addresses questions about nerve and normalization functors in categorified algebra studies.

Contribution

It defines Lie 3-algebras and proves their equivalence with certain truncated Lie infinity algebras, clarifying their structural relationship.

Findings

Lie 3-algebras are in 1-to-1 correspondence with specific 3-term Lie infinity algebras

Addresses the use of nerve and normalization functors in categorified algebra analysis

Provides a new framework for understanding higher categorified algebra structures

Abstract

We define Lie 3-algebras and prove that these are in 1-to-1 correspondence with the 3-term Lie infinity algebras whose bilinear and trilinear maps vanish in degree (1,1) and in total degree 1, respectively. Further, we give an answer to a question of [Roy07] pertaining to the use of the nerve and normalization functors in the study of the relationship between categorified algebras and truncated sh algebras.