The Category of Equivalence Relations

We make some beginning observations about the category 𝔼q of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations R and S is a mapping from the set of R -equivalence classes to that of S -equivalence classes, which is induced by a computable function. We also consider some full subcategories of 𝔼q, such as the category Eq(Σ_1^0) of computably enumerable equivalence relations (called ceers), the category Eq(Π_1^0) of co-computably enumerable equivalence relations, and the category 𝔼q(Dark*) whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in Eq(Σ_1^0) the epimorphisms coincide with the onto morphisms, but in Eq(Π_1^0) there are epimorphisms that are not onto. Moreover, 𝔼q, Eq(Σ_1^0), and 𝔼q(Dark*) are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in Eq(Π_1^0) whose coequalizer in 𝔼q is not an object of Eq(Π_1^0).