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additive quotient category (Definition)

Essential data: Dense subcategory

Definition 0.1   A full subcategory $\mathcal{A}$ of an Abelian category $\mathcal{C}$ is called dense if for any exact sequence in $\mathcal{C}$:

$\displaystyle 0 \to X' \to X \to X'' \to 0,$
$X$ 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 $\mathcal{A}$.

System of morphisms $\Sigma_A$

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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Cross-references: general definition, category, morphisms, system, quotient object, dense subcategory, object, Abelian category

This is version 1 of additive quotient category, born on 2009-05-09.
Object id is 735, canonical name is AdditiveQuotientCategory3.
Accessed 321 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
Pending Errata and Addenda
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