The Category of Equivalence Relations

We make some beginning observations about the category q 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 q, 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 q(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, q, $$ \mathbbm{E}\mathrm{q}\left({\Sigma}_1^0\right), $$ and q(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 q is not an object of $$ \mathbbm{E}\mathrm{q}\left({\Pi}_1^0\right). $$