On the scarcity of contravariant left adjunctions
type: Item
instance of: scholarly article
title: On the scarcity of contravariant left adjunctions
zbMATH Open document ID: 0611.18001
DOI: 10.1007/BF01188394
author: George M. Bergman
published in: Algebra Universalis
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 18A40
Mathematics Subject Classification ID: 18C05
Mathematics Subject Classification ID: 08C05
Mathematics Subject Classification ID: 18A32
zbMATH DE Number: 3987527
zbMATH Keywords: adjoint pairs
zbMATH Keywords: covariant functors
zbMATH Keywords: varieties of algebras
zbMATH Keywords: mutually left adjoint contravariant functors
zbMATH Keywords: infinite product algebra
zbMATH Keywords: quasivarieties
zbMATH Keywords: finitely generated algebras
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01188394
OpenAlex ID: W2081975202
description: scientific article; zbMATH DE number 3987527