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# Mitchellâ€™s embedding theorem

###### Theorem 1.

As a consequence, this theorem says that certain facts about small abelian categories can be proved in the more concrete setting of $\mbox{Mod}_{R}$ (indeed a concrete category). For example, in order to prove that a sequence is exact in an abelian category, it is enough to prove it in the context of $\mbox{Mod}_{R}$, by realizing the fact that objects in $\mbox{Mod}_{R}$ are sets (with structures) and utilizing the elements therein. In particular, the diagram chasing technique popular in homological algebra may be formulated in small abelian categories as a result of this theorem.

# References

- 1
F. Borceux
*Categories and Structures, Handbook of Categorical Algebra II*, Cambridge University Press, Cambridge (1994) - 2
P. Freyd
*Abelian Categories*, Harper and Row, (1964) [online version] - 3
B. Mitchell
*The Full Embedding Theorem*, American Journal of Math, 86, (1964) pp. 619-637

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Freyd-Mitchell embedding theorem

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Theorem

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Reference

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## Mathematics Subject Classification

18E20*no label found*18E10

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