On Crossed Modules in Modified Categories of Interest

TL;DR

This paper explores algebraic structures like singularity, commutators, and central extensions within modified categories of interest, establishing connections to crossed modules and unifying various algebraic notions.

Contribution

It introduces new algebraic structures and links cat$^{1}$-objects to crossed modules in modified categories of interest, unifying multiple concepts.

Findings

Defined singularity, commutators, and central extensions in the new categories.

Connected cat$^{1}$-objects to crossed modules, unifying their theories.

Provided a framework for understanding algebraic structures in modified categories.

Abstract

We introduce some algebraic structures such as singularity, commutators and central extension in modified categories of interest. Additionally, we introduce the cat$^{1}$-objects with their connection to crossed modules in these categories which gives rise to unify many notions about (pre)crossed modules in various algebras of categories.