A note on the category of equivalence relations

TL;DR

This paper explores the categorical properties of equivalence relations on natural numbers, focusing on subcategories defined by computability conditions and analyzing morphism properties and limits within these categories.

Contribution

It introduces and studies the structure of various categories of equivalence relations, highlighting differences in morphism properties and categorical limits among them.

Findings

In $ ext{Eq}( ext{Σ}^0_1)$, epimorphisms are exactly onto morphisms.

In $ ext{Eq}( ext{Π}^0_1)$, some epimorphisms are not onto.

Certain categories are closed under finite limits and colimits, but coequalizers may leave the subcategory.

Abstract

We make some beginning observations about the category $\mathbb{E}\mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ 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 $\mathbb{E}\mathrm{q}$, such as the category $\mathbb{E}\mathrm{q}(\Sigma^0_1)$ of computably enumerable equivalence relations (called ceers), the category $\mathbb{E}\mathrm{q}(\Pi^0_1)$ of co-computably enumerable equivalence relations, and the category $\mathbb{E}\mathrm{q}(\mathrm{Dark}^*)$ 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 $\mathbb{E}\mathrm{q}(\Sigma^0_1)$ the epimorphisms coincide with the onto morphisms, but in $\mathbb{E}\mathrm{q}(\Pi^0_1)$ there are epimorphisms that are not onto. Moreover, $\mathbb{E}\mathrm{q}$, $\mathbb{E}\mathrm{q}(\Sigma^0_1)$, and $\mathbb{E}\mathrm{q}(\mathrm{Dark}^*)$ are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in $\mathbb{E}\mathrm{q}(\Pi^0_1)$ whose coequalizer in $\mathbb{E}\mathrm{q}$ is not an object of $\mathbb{E}\mathrm{q}(\Pi^0_1)$.