The Freyd-Mitchell Embedding Theorem

TL;DR

The paper provides a detailed proof of the Freyd-Mitchell embedding theorem, demonstrating how small abelian categories can be fully embedded into module categories over a ring, facilitating broader algebraic analysis.

Contribution

It offers a comprehensive, step-by-step proof of the embedding theorem, clarifying complex concepts and techniques involved in embedding abelian categories into module categories.

Findings

Proof of the embedding theorem detailed and clarified.

Connections between abelian categories and module categories established.

Conceptual tools like projective generators and injective envelopes explained.

Abstract

Given a small abelian category $\mathcal{A}$, the Freyd-Mitchell embedding theorem states the existence of a ring $R$ and an exact full embedding $\mathcal{A} \rightarrow R$-Mod. This theorem is useful as it allows one to prove general results about abelian categories within the context of $R$-modules. The goal of this report is to flesh out the proof of the embedding theorem. We shall follow closely the material and approach presented in Freyd (1964). This means we will encounter such concepts as projective generators, injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck categories, subcategories of mono objects and subcategories of absolutely pure objects.