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.
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