On multivarieties and multialgebraic categories

Multivarieties are classes of algebras presented by exclusive-orʹs of equations. A full characterization of categories which are equivalent to multivarieties is presented, close to Lawvereʹs characterization of varieties. A comparison with Diers’ concept of multialgebraic category is presented: this is precisely a multivariety with effective equivalence relations. Besides, multialgebraic categories are shown to be precisely those categories which can be sketched by a (finite product, coproduct)-sketch.