On the abstract characterization of quasivarieties (Q1866815)

F. W. Lawvere proved in 1963 that a category is equivalent to a variety of algebras if and only if it is exact and it has an abstractly finite, regular projective, regular generator \(P\). The authors give a characterization of categories which are equivalent to quasivarieties in a similar way: A category \(B\) is equivalent to a quasivariety if and only if \(B\) is a regular category with a finitely presentable, regular projective, regular generator \(P\) and, moreover, it has coequalizers of equivalence relations.