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[parent] examples of abelian categories (Example)

The axiomatization of abelian categories was intended to capture some of the useful properties of categories in homological algebra. This entry gives some examples of abelian categories.

Example 1   The category $ \mathbf{Ab}$ of abelian groups is an abelian category. [Proof]

Since abelian groups are a special case of $ R$-modules, one might ask whether categories of $ R$-modules are also abelian categories. This is the case.

Example 2   For any ring $ R$, the category $ {}_{R}\mathbf{Mod}$ of left $ R$-modules is an abelian category.
Example 3   For any ring $ R$, the category $ \mathcal{C}(R)$ of complexes of left $ R$-modules is an abelian category.
Example 4   For any topological space $ X$, the category of sheaves of abelian groups over $ X$ is an abelian category.



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proof that abelian groups form an abelian category (Proof) by mps
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Cross-references: sheaves, topological space, ring, abelian groups, algebra, categories, abelian categories

This is version 7 of examples of abelian categories, born on 2007-02-20, modified 2007-02-20.
Object id is 8931, canonical name is ExamplesOfAbelianCategory.
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AMS MSC18E10 (Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories)

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