PlanetPhysics/Category of Additive Fractions

Category of Additive Fractions

Let us recall first the necessary [[../PreciseIdea/|concepts]] that enter in the definition of a category of additive fractions.

Dense Subcategory

A full subcategory $𝒜$ of an [[../AbelianCategory2/|abelian category]] $𝒞$ is called dense if for any exact sequence in $𝒞$: $0\to X^{\prime }\to X\to X^{″}\to 0,$ $X$ is in $𝒜$ if and only if both $X^{\prime }$ and $X^{″}$ are in $𝒜$.

Remark 0.1

One can readily prove that if $X$ is an [[../TrivialGroupoid/|object]] of the dense subcategory $𝒜$ of $𝒞$ as defined above, then any subobject $X_Q$, or quotient object of $X$, is also in $𝒜$.

System of morphisms Σ_A

Let $𝒜$ be a dense subcategory (as defined above) of a locally small Abelian category $𝒞$, and let us denote by ${\mathrm{\Sigma }}_A$ (or simply only by $\mathrm{\Sigma }$ -- when there is no possibility of confusion) the [[../GenericityInOpenSystems/|system]] of all [[../TrivialGroupoid/|morphisms]] $s$ of $𝒞$ such that both $kers$ and $cokers$ are in $𝒜$.

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

Quotient Category

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

Remark 0.2

In view of the restriction to additive fractions in the above definition, it may be more appropriate to call the above [[../Cod/|category]] $𝒞/𝒜$ 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 the above remark is also applicable in the context of the more general definition of a quotient category.