# nLab Gabriel multiplication

category theory

## Applications

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Definition

For any two full subcategories $S$ and $T$ of an abelian category $A$, define their Gabriel product $S\bullet T$ as the full subcategory of $A$ generated by all objects $M$ such that $M$ fits in a short exact sequence of the form

$0\to N\to M\to P\to 0$

where $N$ is an object in $S$ and $P$ is an object in $T$.

In the case of the abelian category of modules over a ring, the Gabriel multiplication is sometimes expressed as Gabriel composition of filters of ideals, rather than in terms of abelian subcategories.

## Properties

When restricted to the class of topologizing subcategories, Gabriel multiplication is associative; if $A$ is small then the topologizing subcategories make a semiring with respect to the commutative operation $\cap$ and Gabriel multiplication; in particular $\bullet$ is left and right distributive with respect to intersection of topologizing subcategories.

Revised on May 5, 2011 12:57:52 by Urs Schreiber (89.204.137.116)