The Embedding Theorem

TL;DR

This paper proves that the category of crossed modules over a fixed group is an exact category and provides a complete proof of its embedding into a set-valued functor category.

Contribution

It establishes the exactness of the category of crossed modules and fully proves its embedding theorem into a functor category.

Findings

Category of crossed modules over a fixed group is exact

Complete proof of the embedding theorem into a set-valued functor category

Provides foundational results for categorical algebra

Abstract

In this work, it is shown that the category XMod/P of crossed modules over fixed group P is an exact category and the complete proof of the embedding theorem of XMod/P into a set valued functor category is given.