# nLab subquotient

### Context

#### Limits and colimits

limits and colimits

category theory

# Contents

## Definition

A subquotient of an object $X$ in some category is

or

• a subobject of a quotient object of $X$.

## Properties

• The two definitions are equivalent whenever quotients are stable under pullback and subobjects are stable under pushout, such as in a topos.

• Just as with subobjects and quotient objects, we have that $X$ is a subquotient of itself, and subquotients of subquotients of $X$ are themselves subquotients of $X$ in a natural way.

• Just as subobjects of a set $X$ are in correspondence with predicates on $X$ and quotients of $X$ are in correspondence with equivalence relations on $X$, subquotients of $X$ are in correspondence with partial equivalence relations on $X$.

## Examples

Revised on January 1, 2011 09:33:03 by Urs Schreiber (82.113.106.7)