# nLab quotient object

category theory

## Applications

#### Limits and colimits

limits and colimits

# Contents

## Definition

The quotient object $Q$ of a congruence (an internal equivalence relation) $E$ on an object $X$ in a category $C$ is the coequalizer $Q$ of the induced pair of maps $E \to X$.

If $E$ is additionally the kernel pair of the map $X \to Q$, then $Q$ is called an effective quotient (and $E$ is called an effective congruence, with the map $X \to Q$ being an effective epimorphism).

Sometimes the term is used more loosely to mean an arbitrary coequalizer. It may also refer to a co-subobject of $X$ (that is, a subobject of $X$ in the opposite category $C^\op$), without reference to any congruence on $X$. Note that in a regular category, any regular epimorphism (i.e. a “regular quotient” in the co-subobject sense) is in fact the quotient (= coequalizer) of its kernel pair.

## In higher category theory

These notions have generalizations when $C$ is an (∞,1)-category:

For instance an action groupoid is a quotient of a group action in 2-category theory.

## Examples

Revised on July 13, 2014 03:42:37 by Urs Schreiber (82.113.106.26)