Physics Library
 An open source physics library
Encyclopedia | Forums | Docs | Random | Template Test |  
Login
create new user
Username:
Password:
forget your password?
Main Menu
Sections

Talkback

Downloads

Information
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.

View style:

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 2234 times total.

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

No messages.

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "