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additive quotient category
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(Definition)
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Definition 0.1 A full subcategory $\mathcal{A}$ of an Abelian category $\mathcal{C}$ is called dense if for any exact sequence in $\mathcal{C}$ : $$ 0 \to X' \to X \to X'' \to 0,$$ $X$ is in $\mathcal{A}$ if and only if both $X'$ and $X''$ are in
$\mathcal{A}$ .
Remark 0.1: One can readily prove that if $X$ is an object of the dense subcategory $\mathcal{A}$ of $\mathcal{C}$ as defined above, then any subobject $X_Q$ , or quotient object of $X$ , is also in $\mathcal{A}$ .
Let $\mathcal{A}$ be a dense subcategory (as defined above) of a locally small Abelian category $\mathcal{C}$ , and let us denote by $\Sigma_A$ (or simply only by $\Sigma$ - when there is no possibility of confusion) the system of all morphisms $s$ of $\mathcal{C}$ such that both $ker s$ and $coker s$ are in $\mathcal{A}$ . One can then prove that the category of additive fractions $\mathcal{C}_{\Sigma}$ of $\mathcal{C}$ relative to $\Sigma$ exists.
Definition 0.2 The quotient category of $\mathcal{C}$ relative to $\mathcal{A}$ , denoted as $\mathcal{C}/\mathcal{A}$ , is defined as the category of additive fractions $\mathcal{C}_{\Sigma}$ relative to a class of morphisms $\Sigma :=\Sigma_A $ in $\mathcal{C}$ .
Remark 0.2 In view of the restriction to additive fractions in the above definition, it may be more appropriate to call the above category $\mathcal{C}/\mathcal{A}$ an additive quotient category. This would be important in order to avoid confusion with the more general notion of quotient category-which is defined as a category of fractions. Note however that Remark 0.1 is also applicable in the context of the more general definition of a quotient category.
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"additive quotient category" is owned by bci1.
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Cross-references: category of fractions, order, category, fractions, additive, restriction, class, quotient category, category of additive fractions, morphisms, locally small, quotient object, subobject, object, exact sequence, dense, abelian category, full subcategory
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This is version 28 of additive quotient category, born on 2008-10-08, modified 2008-10-27.
Object id is 11161, canonical name is QuotientCategory.
Accessed 1204 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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