PlanetPhysics/Additive Quotient Category 3

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Essential data: Dense subcategory[edit | edit source]

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 ΣA[edit | edit source]

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.