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

### 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 $\Sigma _{A}$ 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 $kers$  and $cokers$  are in ${\mathcal {A}}$ .

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

### Quotient Category

A quotient category of $\displaystyle \mathcal{C$ 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.