Geodesic
Characterizations of Axiomatic Categories of Models Canonically Isomorphic to (Quasi-)Varieties
Canadian mathematical bulletin, Tome 31 (1988) no. 3, pp. 287-300

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DOI

Let be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if 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 of L which 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)”.
DOI : 10.4153/CMB-1988-042-x
Mots-clés : Categories of models, preservation theorem, varieties, monadicity, 03C60, 08C10, 18C15 
Hébert, Michel. Characterizations of Axiomatic Categories of Models Canonically Isomorphic to (Quasi-)Varieties. Canadian mathematical bulletin, Tome 31 (1988) no. 3, pp. 287-300. doi: 10.4153/CMB-1988-042-x
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     title = {Characterizations of {Axiomatic} {Categories} of {Models} {Canonically} {Isomorphic} to {(Quasi-)Varieties}},
     journal = {Canadian mathematical bulletin},
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     year = {1988},
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     doi = {10.4153/CMB-1988-042-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1988-042-x/}
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