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 $\mathbbm{E}\mathrm{q}\left({\Sigma}_1^0\right)$ of computably enumerable equivalence relations (called ceers), the category $\mathbbm{E}\mathrm{q}\left({\Pi}_1^0\right)$ 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 $\mathbbm{E}\mathrm{q}\left({\Sigma}_1^0\right)$ the epimorphisms coincide with the onto morphisms, but in $\mathbbm{E}\mathrm{q}\left({\Pi}_1^0\right)$ there are epimorphisms that are not onto. Moreover, 𝔼q, $\mathbbm{E}\mathrm{q}\left({\Sigma}_1^0\right),$ and 𝔼q(Dark*) are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in $\mathbbm{E}\mathrm{q}\left({\Pi}_1^0\right)$ whose coequalizer in 𝔼q is not an object of $\mathbbm{E}\mathrm{q}\left({\Pi}_1^0\right).$