PlanetPhysics/Additive Quotient Category 3

Essential data: 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 Σ_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 ${\mathcal {C}}_{\Sigma }$ of ${\mathcal {C}}$ relative to $\Sigma$ exists.

The quotient category of Failed to parse (syntax error): {\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 Remark 0.1 is also applicable in the context of the more general definition of a quotient category.