nLab
congruence

Definitions

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

To be precise, this consists of a subobject $R\stackrel{(p_1,p_2)}{\hookrightarrow }X\times X$ equipped with the following maps:

• 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

  $$\begin{array}{ccc}R{\times }_{X}R& \stackrel{q_2}{\to }& R\\ {\downarrow }^{q_1}& & {\downarrow }^{p_1}\\ R& \stackrel{p_2}{\to }& X\end{array}$$\array{ R \times_X R & \stackrel{q_2}\rightarrow & R\\ \downarrow^{q_1} && \downarrow^{p_1}\\ R & \stackrel{p_2}\rightarrow & X }

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

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

Any kernel pair is a congruence; a congruence which is the kernel pair of some morphism is called effective. The coequalizer of a congruence is called a quotient object. An effective congruence is always the kernel pair of its quotient if that quotient exists; the quotient of an effective congruence is an effective quotient. A regular category is called an exact category if every congruence is effective.

Examples

• 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.

Related pages

• 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.