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Gabriel-Popescu theorem for  -categories
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(Theorem)
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Theorem 0.1 Let $\mathcal{A}$ be an $\mathcal{A}b5$ -category, $U$ an object of $\mathcal{A}$ , and $$A = End_{\mathcal{A}}(U)$$ ; also, let $S: \mathcal{A} \to Mod A^0$ be the functor defined by $$S(X) = Hom_{\mathcal{A}} (U,X),$$ and $T$ its left adjoint. Then, the following two statements are equivalent:
- $U$ is a generator of $\mathcal{A}$ ;
- $S$ is full and faithful and $T$ is exact.
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"Gabriel-Popescu theorem for  -categories" is owned by bci1.
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Cross-references: faithful, generator, equivalent, left adjoint, functor, object
This is version 4 of Gabriel-Popescu theorem for  -categories, born on 2008-10-04, modified 2009-01-29.
Object id is 11133, canonical name is GabrielPopescuTheoremForAb5Categories.
Accessed 517 times total.
Classification:
| AMS MSC: | 18-00 (Category theory; homological algebra :: General reference works ) | | | 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories) |
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Pending Errata and Addenda
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