Characterizations of Axiomatic Categories of Models Canonically Isomorphic to (Quasi-)Varieties
Michel Hébert · Canadian Mathematical Bulletin · 1988
Abstract Let be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentencesTin a finitary first-order languageL.Functors between two such categories are said to becanonicalif they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion ofLwhich assigns to a model its (unique) expansion. This solves a problem of H. Volger. In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.