Congruences

category theory

# Congruences

## Definition

In a finitely complete category $C$, a setoid object is an object $X$ with an internal pseudo-equivalence relation on $X$. This means that it consists of a an object $X$, an object $R$, and morphisms $p_1:R \to X$ and $p_2:R \to X$, 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

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