Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
In a category $C$ with pullbacks, an internal pseudo-equivalence relation on an object $X$ is an object $R$ with morphisms $p_1:R \to X$ and $p_2:R \to X$, along 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
An object with an internal pseudo-equivalence relation is sometimes called a setoid object, but those are also used for objects with an internal equivalence relation; i.e. a congruence.
Last revised on June 22, 2023 at 13:59:46. See the history of this page for a list of all contributions to it.