Fredom-Mitchell嵌入定理
The Freyd-Mitchell Embedding Theorem
摘要：
给定一个小的阿贝尔范畴$\mathcal{A}$，Frederic-Mitchell嵌入定理陈述了环$R$和精确的全嵌入$\mathcal{A} \rightarrow R$ -模的存在性。这个定理是有用的，因为它允许人们在$R$ -模的上下文中证明关于阿贝尔范畴的一般结果。本报告的目的是充实嵌入定理的证明。我们将密切关注Freyd（1964）中提出的材料和方法。这意味着我们将遇到诸如投射生成元、内射余生成元、米田嵌入、内射包络、格罗滕迪克范畴、单对象的子范畴和绝对纯对象的子范畴等概念。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
