Categorification of Algebras: 2-Algebras

TL;DR

This paper develops a new framework called 2-algebras that categorifies k-algebras using 2-categories, establishing their equivalence to crossed modules and exploring homotopy relations.

Contribution

It introduces 2-algebras as a categorification of k-algebras, linking them to crossed modules and defining homotopy for these structures.

Findings

Category of 2-algebras is equivalent to crossed modules in commutative k-algebras.

Defined homotopy for 2-algebras and related it to crossed module homotopy.

Established foundational properties of 2-algebras and their homotopies.

Abstract

This paper introduces a categorification of $k$-algebras called 2 -algebras, where k is a commutative ring. We define the 2-algebras as a 2-category with single object in which collections of all 1-morphisms and all 2-morphisms are k-algebras. It is shown that the category of 2 -algebras is equivalent to the category of crossed modules in commutative k -algebras. Also we define the notion of homotopy for 2-algebras and we explore the relations of crossed module homotopy and 2-algebra homotopy.