Congruences

category theory

# Congruences

## Idea

In Euclidean geometry, by congruence one means the equivalence relation on the collection of subsets of a Euclidean space which regards two of these as equivalent if one is carried into the other by an isometry (of the ambient Euclidean space).

Similarly, in algebra, by a congruence one means certain equivalence relations on elements of algebraic structures, such as groups or rings (cf. e.g. the multiplicative group of integers modulo n).

Therefore, in category theory the term congruence is used in the broad generality of equivalence relations on (the generalized elements of) any object internal to any finitely complete category.

## Definitions

###### Definition

In a category $C$ with pullbacks, a congruence on an object $X$ is an internal equivalence relation on $X$ (i.e.: an internal groupoid — hence an internal category with all morphisms being isomorphisms — but with no non-identity automorphisms).

This means that it consists of a subobject

(1)$i \;\colon\; R\stackrel{(p_1,p_2)}\hookrightarrow X \times X$

of the Cartesian product of $X$ with itself, 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:

$t \,\colon\, R \times_X R \to R$

(where on the left we have the fiber product of $R \hookrightarrow X \times X \overset{p_2}{\to} X$ with $R \hookrightarrow X \times X \overset{p_1}{\to} X$, i.e. the subobject of pairs of composable pairs in relation)

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

$\array{ && R \\ & {}^{\mathllap{t}}\nearrow & \big\downarrow \\ R \times_X R & \underset{(p_1 q_1,p_2 q_2)}{\longrightarrow} & X \times X \mathrlap{\,,} }$

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

$\array{ R \times_X R & \overset{q_2}\longrightarrow & R \\ \big\downarrow{\mathrlap{{}^{q_1}}} && \big\downarrow{\mathrlap{{}^{p_1}}} \\ R & \underset{p_2}\longrightarrow & X }$
###### 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.

## References

### In category theory

References using terminology as above:

But, for what it’s worth, a different use of the term “congruence” in category theory appears in Def. 3.5.1 on p. 89 in:

• Michael Barr, Charles Wells, Category theory for computing science, Prentice-Hall International Series in Computer Science (1995); reprinted in: Reprints in Theory and Applications of Categories 22 (2012) 1-538 [pdf, tac:tr22]

Last revised on July 30, 2024 at 13:47:01. See the history of this page for a list of all contributions to it.