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\begin{document}

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\section*{Freyd-Mitchell embedding theorem}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{additive_and_abelian_categories}{}\paragraph*{{Additive and abelian categories}}\label{additive_and_abelian_categories}

[[!include additive and abelian categories - contents]]

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Freyd--Mitchell embedding theorem} says that every [[abelian category]] is a [[full subcategory]] of a [[category of modules]] over some [[ring]] $R$ and that the embedding is an [[exact functor]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\begin{remark}
\label{}\hypertarget{}{}
It is easy to see that not every abelian category is \emph{[[equivalence of categories|equivalent]]} to $R$[[Mod]] for some [[ring]] $R$. The reason is that $R Mod$ has all [[small category|small]] [[limits]] and [[colimits]]. But for instance, for $R$ [[noetherian ring|Noetherian]], the category of [[finitely generated module|finitely generated]] $R$-modules is an abelian category but lacks these properties.

\end{remark}
However, we have

\begin{theorem}
\label{}\hypertarget{}{}
Every [[small category|small]] [[abelian category]] admits a [[full functor|full]], [[faithful functor|faithful]] and [[exact functor|exact]] functor to the category $R$[[Mod]] for some [[ring]] $R$.

\end{theorem}
This result can be found as Theorem 7.34 on page 150 of (\hyperlink{Freyd}{Freyd}). (The terminology there is a bit outdated, in that it calls an abelian category ``fully abelian'' if it admits a full and faithful exact functor to a category of $R$-modules.) A pedagogical discussion is in section 1.6 of (\hyperlink{Weibel}{Weibel}). See also (\hyperlink{Wikipedia}{Wikipedia}) for the idea of the proof.

\begin{proof}
(\ldots{})

\end{proof}
We can also characterize which abelian categories \emph{are} equivalent to a category of $R$-modules:

\begin{theorem}
\label{}\hypertarget{}{}
Let $C$ be an [[abelian category]]. If $C$ has all [[small category|small]] [[coproducts]] and has a [[compact object|compact]] [[projective object|projective]] [[generator]], then $C \simeq R Mod$ for some ring $R$.

In fact, in this situation we can take $R = C(x,x)^{op}$ where $x$ is any compact projective generator. Conversely, if $C \simeq R Mod$, then $C$ has all small coproducts and $x = R$ is a compact projective generator.

\end{theorem}
This theorem, minus the explicit description of $R$, can be found as Exercise F on page 103 of (\hyperlink{Freyd}{Freyd}). The first part of this theorem can also be found as Prop. 2.1.7 in (\hyperlink{Ginzburg}{Ginzburg}). Conversely, it is easy to see that $R$ is a compact projective generator of $R Mod$.

Going further, we can try to characterize functors between categories of $R$-modules that come from tensoring with bimodules. Here we have

\begin{theorem}
\label{}\hypertarget{}{}
If $B$ is an an $S$-$R$-bimodule, the [[tensor product]] functor

\begin{displaymath}
B \otimes_R -\colon R Mod \to S Mod
\end{displaymath}
is [[right exact functor|right exact]] and preserves [[small category|small]] [[coproducts]]. Conversely, if $F\colon Mod_R \to Mod_S$ is right exact and that preserves small coproducts, it is naturally isomorphic to $B \otimes_R -$ where $B$ is the $S$-$R$-bimodule $F R$.

\end{theorem}
This theorem was more or less simultaneously proved by Watts and Eilenberg; a generalization is proved in (\hyperlink{NymanSmith}{Nyman-Smith}), and references to the original papers can be found there.

Going still further we should be able to obtain a nice theorem describing the [[image]] of the embedding of the [[2-category]] of

\begin{itemize}%
\item rings
\item bimodules
\item bimodule homomorphisms

\end{itemize}
into the strict 2-category of

\begin{itemize}%
\item abelian categories
\item right exact functors
\item natural transformations.

\end{itemize}
For more discussion see the \href{http://golem.ph.utexas.edu/category/2007/08/questions_about_modules.html}{$n$-Cafe}.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \href{stable+model+category#AsCategoriesOfModules}{stable model category -- as A-infinity algebroid module categories}

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A standard textbook is

\begin{itemize}%
\item [[Peter Freyd]], \emph{\href{http://www.tac.mta.ca/tac/reprints/articles/3/tr3.pdf}{Abelian Categories (pdf)})}

\end{itemize}
Details on the proof and its variants are also in

section 1.6 of

\begin{itemize}%
\item [[Charles Weibel]], \emph{[[An Introduction to Homological Algebra]]}

\end{itemize}
and

\begin{itemize}%
\item [[Victor Ginzburg]], \emph{Lectures on noncommutative geometry} (\href{http://arxiv.org/PS_cache/math/pdf/0506/0506603v1.pdf}{pdf})

\end{itemize}
\begin{itemize}%
\item A. Nyman , S. Paul Smith, \emph{A generalization of Watts's Theorem: Right exact functors on module categories} (\href{http://arxiv.org/abs/0806.0832}{arXiv:0806.0832})

\end{itemize}
An introductory survey is for instance also in section 3 of

\begin{itemize}%
\item Geillan Aly, \emph{Abelian Categories and the Freyd-Mitchell Embedding Theorem} (\href{http://www.u.arizona.edu/~geillan/research/ab_categories.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem}{Mitchell's embedding theorem}}

\end{itemize}
[[!redirects Freyd-Mitchell embedding theorem]] [[!redirects Freyd–Mitchell embedding theorem]] [[!redirects Freyd--Mitchell embedding theorem]] [[!redirects Freyd-Mitchell embedding]] [[!redirects Freyd–Mitchell embedding]] [[!redirects Freyd--Mitchell embedding]] [[!redirects Freyd-Mitchell theorem]] [[!redirects Freyd–Mitchell theorem]] [[!redirects Freyd--Mitchell theorem]]

[[!redirects Mitchell theorem]] [[!redirects Mitchell's theorem]] [[!redirects Mitchell's theorem]]

[[!redirects Watt's theorem]] [[!redirects Watt theorem]]



\end{document}