PlanetPhysics/Additive Quotient Category 3

Essential data: Dense subcategory
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 $$\mathcal{A}$$. \\

System of morphisms ΣA
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.

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.