Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
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
where $q_1$ and $q_2$ are the projections defined by the pullback diagram
Since $i$ is a monomorphism, the maps $r$, $s$, and $t$ are necessarily unique if they exist.
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).
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.
The coequalizer of a congruence is called a quotient object.
The quotient of an effective congruence is called an effective quotient.
A regular category is called an exact category if every congruence is effective.
An effective congruence, def. , is always the kernel pair of its quotient, def. , if that quotient exists.
Every kernel pair is a congruence.
An equivalence relation is precisely a congruence in Set.
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.
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.
The notions of regular category and exact category can naturally be formulated in terms of congruences. A “higher arity” version, corresponding to coherent categories and pretoposes is discussed at familial regularity and exactness.
A Mal'cev category is a finitely complete category in which every internal relation satisfying reflexivity is thereby actually a congruence.
Higher-categorical generalizations are that of a 2-congruence and of a groupoid object in an (∞,1)-category. See also (n,r)-congruence.
Last revised on April 2, 2021 at 19:28:18. See the history of this page for a list of all contributions to it.