# PlanetPhysics/Additive Quotient Category 3

### Essential data: 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 0.1:**
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 of
relative to exists.

The 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 0.2**
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 **Remark 0.1** is also applicable in the context of the more general definition of a quotient category.