Hébert, Michel Sur les algèbres libres pour les classes axiomatiques. (On free algebras for axiomatic classes). (French) Zbl 0689.08005 Ann. Sci. Math. Qué. 13, No. 1, 39-47 (1989). Let T be a finitary algebraic theory and let K be a class of T-algebras (considered as a full subcategory of the category of T-algebras and T- homomorphisms). This paper considers questions concerning free algebra constructions for K. There is the universal algebraic approach of the free algebra as being the algebra generated by the set on which it is free. This free algebra is formally constructed as a quotient of the free T-structure of terms. On the other hand, there is the category theoretical point of view dealing with the existence of the left adjoint to the forgetful functor U: \(K\to Sets\) and the corresponding universal property. The properties of these constructions are compared and analyzed with regard to infinite underlying sets of various cardinalities. The discussion involves considerations of things such as the existence of a codensity monad for U of finite rank. Reviewer: K.I.Rosenthal MSC: 08B20 Free algebras 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 08C05 Categories of algebras Keywords:finitary algebraic theory; free algebra constructions; left adjoint to the forgetful functor; codensity monad PDFBibTeX XMLCite \textit{M. Hébert}, Ann. Sci. Math. Qué. 13, No. 1, 39--47 (1989; Zbl 0689.08005)