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Mitchell's embedding theorem
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(Theorem)
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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.
- 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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"Mitchell's embedding theorem" is owned by CWoo.
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Freyd-Mitchell embedding theorem |
This object's parent.
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Cross-references: algebra, diagram, structures, objects, sequence, order, concrete category, theorem, consequence, ring, modules, category, abelian category
There is 1 reference to this entry.
This is version 4 of Mitchell's embedding theorem, born on 2008-10-16, modified 2008-10-17.
Object id is 11180, canonical name is MitchellsEmbeddingTheorem.
Accessed 661 times total.
Classification:
| AMS MSC: | 18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories) | | | 18E20 (Category theory; homological algebra :: Abelian categories :: Embedding theorems) |
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Pending Errata and Addenda
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