On quasi-equations in locally presentable categories II: a logic

Abstract

Quasi-equations given by parallel pairs of finitary morphisms represent properties of objects: an object satisfies the property if its contravariant homfunctor merges the parallel pair. Recently Ad´amek and H´ebert characterized subcategories of locally finitely presentable categories specified by quasi-equations. We now present a logic of quasi-equations close to Birkhoff’s classical equational logic. We prove that it is complete in all locally finitely presentable categories with effective equivalence relations.