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category of additive fractions (Topic)

Category of Additive Fractions

Let us recall first the necessary concepts that enter in the definition of a category of additive fractions.

Dense Subcategory

Definition 1.1   A full subcategory $\mathcal{A}$ of an Abelian category $\mathcal{C}$ is called dense if for any exact sequence in $\mathcal{C}$:

$\displaystyle 0 \to X' \to X \to X'' \to 0,$
$X$ 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 $\Sigma_A$

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 $ker s$ and $coker s$ are in $\mathcal{A}$.

One can then prove that the category of additive fractions $\mathcal{C}_{\Sigma}$ of $\mathcal{C}$ relative to $\Sigma$ exists.

Quotient Category

Definition 1.2   A quotient category of $\mathcal{C}$ 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 the above remark is also applicable in the context of the more general definition of a quotient category.



"category of additive fractions" is owned by bci1.

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Other names:  quotient category, additive quotient category, additive category
Also defines:  quotient object, quotient category, additive quotient category, category of additive fractions, dense subcategory, system of morphisms
Keywords:  category of additive fractions, additive category, additive quotient category

Cross-references: category, morphisms, system, object, Abelian category, concepts
There are 14 references to this object.

This is version 9 of category of additive fractions, born on 2009-02-13, modified 2009-02-13.
Object id is 518, canonical name is CategoryOfAdditiveFractions.
Accessed 2223 times total.

Classification:
Physics Classification: 00. (GENERAL)
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
None.
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