The factor set of Gr-categories of the type $(\Pi,A)$

TL;DR

This paper investigates the structure of $ ext{Gr}$-categories graded by a group $ ext{Gamma}$, focusing on factor sets and their cohomological classification, providing new insights into categorical group theory.

Contribution

It introduces a cohomological classification theorem for $ ext{Gamma}$-graded categorical groups using factor sets and $ ext{Gamma}$-operator 3-cocycles.

Findings

Established a cohomological classification for $ ext{Gamma}$-graded Gr-categories.

Interpreted $ ext{Gamma}$-operator 3-cocycles within categorical group structures.

Connected factor sets with cohomology theory in categorical groups.

Abstract

Any $\Gamma$-graded categorical group is determined by a factor set of a categorical group. This paper studies the factor set of the group $\Gamma$ with coefficients in the categorical group of the type $(\Pi,A).$ Then, an interpretation of the notion of $\Gamma-$operator $3-$cocycle is presented and the proof of cohomological classification theorem for the a $\Gamma-$graded Gr-category is also presented.