# PlanetPhysics/Category of Additive Fractions

## Category of Additive FractionsEdit

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

### Dense SubcategoryEdit

A full subcategory of an abelian category is called *dense* if for any exact sequence in :
is in if and only if both and are in .

### Remark 0.1Edit

One can readily prove that if is an object of the *dense subcategory* of
as defined above, then any subobject , or quotient object of , is also in
.

### System of morphisms Edit

Let be a *dense subcategory* (as defined above) of a locally small Abelian category ,
and let us denote by (or simply only by -- when there is no possibility of confusion) the system of all morphisms of such that both and are in .

One can then prove that the *category of additive fractions* **Failed to parse (syntax error): {\displaystyle \mathcal{C _{\Sigma}}**
of relative to } exists.

### Quotient CategoryEdit

A quotient category of **Failed to parse (syntax error): {\displaystyle \mathcal{C }**
relative to }, denoted as , is defined as the category of additive fractions relative to a class of morphisms
in .

#### Remark 0.2Edit

In view of the restriction to additive fractions in the above definition, it may be more appropriate to call the above category 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.