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

Dense Subcategory

A full subcategory ${\displaystyle {\mathcal {A}}}$  of an abelian category ${\displaystyle {\mathcal {C}}}$  is called dense if for any exact sequence in ${\displaystyle {\mathcal {C}}}$ : ${\displaystyle 0\to X'\to X\to X''\to 0,}$  ${\displaystyle X}$  is in ${\displaystyle {\mathcal {A}}}$  if and only if both ${\displaystyle X'}$  and ${\displaystyle X''}$  are in ${\displaystyle {\mathcal {A}}}$ .

Remark 0.1

One can readily prove that if ${\displaystyle X}$  is an object of the dense subcategory ${\displaystyle {\mathcal {A}}}$  of ${\displaystyle {\mathcal {C}}}$  as defined above, then any subobject ${\displaystyle X_{Q}}$ , or quotient object of ${\displaystyle X}$ , is also in ${\displaystyle {\mathcal {A}}}$ .

System of morphisms ${\displaystyle \Sigma _{A}}$

Let ${\displaystyle {\mathcal {A}}}$  be a dense subcategory (as defined above) of a locally small Abelian category ${\displaystyle {\mathcal {C}}}$ , and let us denote by ${\displaystyle \Sigma _{A}}$  (or simply only by ${\displaystyle \Sigma }$  -- when there is no possibility of confusion) the system of all morphisms ${\displaystyle s}$  of ${\displaystyle {\mathcal {C}}}$  such that both ${\displaystyle kers}$  and ${\displaystyle cokers}$  are in ${\displaystyle {\mathcal {A}}}$ .

One can then prove that the category of additive fractions $\displaystyle \mathcal{C _{\Sigma}$ of ${\displaystyle {\mathcal {C}}}$  relative to ${\displaystyle \Sigma }$ } exists.

Quotient Category

A quotient category of $\displaystyle \mathcal{C$ relative to ${\displaystyle {\mathcal {A}}}$ }, denoted as ${\displaystyle {\mathcal {C}}/{\mathcal {A}}}$ , is defined as the category of additive fractions ${\displaystyle {\mathcal {C}}_{\Sigma }}$  relative to a class of morphisms ${\displaystyle \Sigma :=\Sigma _{A}}$  in ${\displaystyle {\mathcal {C}}}$ .

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 ${\displaystyle {\mathcal {C}}/{\mathcal {A}}}$  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.