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.