Characterizations of axiomatic categories of models canonically isomorphic to (quasi-)varieties

Michel Hébert
1988 Canadian mathematical bulletin  
Let Jt L (T) 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 oiJt L {T) for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is
more » ... a canonical isomorphism from Jt L {T} 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: "^# L (7) is canonically isomorphic to a finitary variety (resp. quasivariety)" and, for the variety case, to "the forgetful functor oiJt L (T) is monadic (tripleable)".
doi:10.4153/cmb-1988-042-x fatcat:osanfbd45vhbnnlz27buf7omye