## You are here

Homequotient category, additive

## Primary tabs

# quotient category, additive

# 0.1 Essential data: Dense subcategory

###### Definition 0.1.

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

$0\to X^{{\prime}}\to X\to X^{{\prime\prime}}\to 0,$ |

$X$ is in $\mathcal{A}$ if and only if both $X^{{\prime}}$ and $X^{{\prime\prime}}$ 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
$\mathcal{A}$.

# 0.1.1 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 $kers$ 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.

###### Definition 0.2.

The *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 Remark 0.1 is also applicable in the context of the more general definition of a quotient category.

## Mathematics Subject Classification

18E05*no label found*18-00

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections