| Abstract: | Abstract The well known characterizations of equational classes of algebras with not necessaryly finitary operations by FELSCHER 6.7] and of categories of A-algebras for algebraic theories A in the sense of LINTON 10], esp., by means of their forgetful functors are the foundations of a concept of varietal functors U:K → L over arbitrary basecategories L. They prove to be monadic functors which satisfy an additional HOM-condition 17]. (In the case L = Set this condition is always fulfilled, see LINTON 11].) Contrary to monadic functors, varietal functors are closed under composition. Pleasent algebraic properties of the base-category L can be ‘lifted’ along varietal functors, such as e.g. factorization properties, (co-) completeness, classical isomorphism theorems, etc. By means of the well known EILENBERG-MOORE-algebras there is a universal monadic functor UT:L T → L for any functor U: K → L, having a left adjoint F (T: = UF). But, in general, UT is not varietal. Under some suitable conditions, however it is possible, to construct a canonical varietal functor ?:R → L, the varietal hull of U. This hull has much more interesting (algebraic) properties than the EILENBERG-MOORE construction. Moreover, results of BANASCHEWSKI-HERRLICH 2] are extended. |
