Congruences

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 of the product $R\stackrel{(p_1,p_2)}\hookrightarrow X \times X$ equipped with the following morphisms:

• internal reflexivity: $r \colon X \to R$ which is a section both of $p_1$ and of $p_2$, i.e., $p_1 r = p_2 r = 1_X$;

• internal symmetry: $s \colon R \to R$ which interchanges $p_1$ and $p_2$, i.e., $p_1\circ s = p_2$ and $p_2\circ s = p_1$;

• internal transitivity: $t: R \times_X R \to R$ which factors the left/right projection map $R \times_X R \to X \times X$ through $R$, i.e., the following diagram commutes

$\array{ && R \\ & {}^{\mathllap{t}}\nearrow & \downarrow \\ R \times_X R & \stackrel{(p_1 q_1,p_2 q_2)}\rightarrow & X \times X }$

where $q_1$ and $q_2$ are the projections defined by the pullback diagram

$\array{ R \times_X R & \stackrel{q_2}\rightarrow & R \\ \downarrow^{\mathrlap{q_1}} && \downarrow^{\mathrlap{p_1}} \\ R & \stackrel{p_2}\rightarrow & X }$
###### Remark

Since $i$ is a monomorphism, the maps $r$, $s$, and $t$ are necessarily unique if they exist.

###### Remark

Equivalently, a congruence on $X$ is an internal category with $X$ the object of objects, such that the (source,target)-map is a monomorphism and such that if there is a morphism $x_1 \to x_2$ then there is also a morphism $x_2 \to x_1$ (internally).

###### Remark

We can equivalently define a congruence $R$ as (a representing object of) a representable sub-presheaf of $\hom(-, X \times X)$ so that for each object $Y$, the composite of $R(Y) \hookrightarrow \hom(Y, X \times X) \cong \hom(Y, X) \times \hom(Y, X)$ exhibits $R(Y)$ as an equivalence relation on the set $\hom(Y, X)$. The upshot of this definition is that it makes sense even when $C$ is not finitely complete.

###### Definition

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

###### Definition

The coequalizer of a congruence is called a quotient object.

The quotient of an effective congruence is called an effective quotient.

###### Definition

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

## Properties

###### Proposition

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

## Examples

###### Example

Every diagonal morphism on an object is a congruence and has a quotient object isomorphic to the original object.

###### Example

Every kernel pair is a congruence.

###### 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 congruence $R = \{(x,y) : xy^{-1} \in K \}$.

Alternatively, a quotient group by a normal subgroup $K \hookrightarrow G$ is the quotient of the congruence $G \times K \stackrel{(p_1,p_2)}{\hookrightarrow} G \times G$, 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.

Last revised on June 12, 2021 at 16:46:50. See the history of this page for a list of all contributions to it.