On the scarcity of contravariant left adjunctions
Although adjoint pairs of covariant functors (one left adjoint and one right adjoint) between varieties of algebras, and pairs of mutually right adjoint contravariant functors, are both familiar entities, it is shown in this note that any pair of mutually left adjoint contravariant functors between varieties of algebras C and D must be fairly degenerate; specifically, it must factor through the retractions of these categories onto the complete lattices of epimorphs of their respective initial objects. The proof is based on the observation that a left adjoint contravariant functor turns limits into colimits, and in particular, products into coproducts, but that the canonical morphism from an infinite product algebra \(\prod _{I}A_ i\) to the reduced product \(\prod _{F}A_ i\) with respect to the filter F of cofinal subsets of the index-set I has properties which cannot be duplicated for a morphism into a coproduct algebra, unless this morphism is quite degenerate. The result is first proved for varieties of finitary algebras, then generalized to possibly infinitary algebras, and to quasivarieties as well as varieties. On the other hand, some examples of contravariant left adjoint functors between the full subcategories of finitely generated or otherwise restricted algebras in certain varieties (e.g., the duality of finite- dimensional vector-spaces over a field, duality of finite abelian groups, and the duality between finite sets and finite Boolean rings) are examined.
mutually left adjoint contravariant functors
scientific article; zbMATH DE number 3987527