Algebraic Theories of Quasivarieties

Abstract

Analogously to the fact that Lawvere's algebraic theories of (finitary) varieties are precisely the small categories with finite products, we prove that (i) algebraic theories of many-sorted quasivarieties are precisely the small, left exact categories with enough regular injectives and (ii) algebraic theories of many-sorted Horn classes are precisely the small left exact categories with enough M-injectives, where M is a class of monomorphisms closed under finite products and containing all regular monomorphisms. We also present a Gabriel–Ulmer-type duality theory for quasivarieties and Horn classes.