category of additive fractions

Category of Additive Fractions

Let us recall first the necessary concepts that enter in the definition of a category of additive fractions.

Dense Subcategory

Definition 1.1   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.

Quotient Category

Definition 1.2   A 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 the above remark is also applicable in the context of the more general definition of a quotient category.



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