category theory

# Congruences

## Definitions

###### Definition

In a finitely complete category $C$, a congruence on an object $X$ is an internal equivalence relation on $X$.

This means that it consists of a subobject $R\stackrel{(p_1,p_2)}\hookrightarrow X \times X$ equipped with the following morphisms:

• internal reflexivity: $r: X \to R$ which is a section both of $p_1$ and of $p_2$;
• internal symmetry: $s: R \to R$ which interchanges $p_1$ and $p_2$, namely $p_1\circ s = p_2$ and $p_2\circ s = p_1$;
• internal transitivity: $t: R \times_X R \to R$; where with the notation for the projections in the cartesian square
$\array{ R \times_X R & \stackrel{q_2}\rightarrow & R\\ \downarrow^{q_1} && \downarrow^{p_2}\\ R & \stackrel{p_1}\rightarrow & X }$

the following holds: $p_1\circ q_1 = p_1\circ t$ and $p_2\circ q_2 = p_2\circ t$.

###### Remark

Since $(p_1,p_2)$ is a monomorphism, the maps $r$, $s$, and $t$ are necessarily unique if they exist.

###### Example

Every kernel pair is a congruence.

###### Definition

A congruence which is the kernel pair of some morphism is called effective.

###### Definition

The coequalizer of a congruence is called a quotient object.

###### Proposition

An effective congruence is always the kernel pair of its quotient if that quotient exists.

###### Definition

The quotient of an effective congruence is an effective quotient.

###### Definition

A regular category is called an exact category if every congruence is effective.

## Examples

###### Example

An equivalence relation is precisely a congruence in Set.

###### Example

The eponymous example is congruence modulo $n$ (for a fixed natural number $n$), which can be considered a congruence on $\mathbb{N}$ in the category of rigs, or on $\mathbb{Z}$ in the category of rings.

###### Example

A quotient group by a normal subgroup $K \hookrightarrow G$ is the quotient of the relation $G \times K \stackrel{(p_1,p_2)}{\hookrightarrow} G \times G$, where $p_1$ is projection on the first factor and $p_2$ is multiplication in $G$ (these are source and target maps in the action groupoid $G \sslash K$).

A special case of this is that of a quotient module.

Revised on September 11, 2012 10:10:02 by Urs Schreiber (82.169.65.155)