Dual adjunctions between enriched algebraic categories

Abstract

Working in the setting of enriched algebraic theories for a system of arities, we study several aspects of dual adjunctions between enriched algebraic categories, which we call algebraic dual adjunctions. Firstly, we generalize Freyd's theorem on contravariant algebra-valued right-adjoint functors to this setting. Secondly, we establish a biequivalence between a 2-category of algebraic dual adjunctions and a locally discrete 2-category of bifold algebras, in a sense defined in prior work of the author. Thirdly, we define special classes of algebraic dual adjunctions that we call (left- and right-)stable, in which certain free objects are reflexive, and we establish biequivalences between these and special classes of bifold algebras defined in terms of commutants in prior work of the author. Fourthly, we show that every algebra for an enriched algebraic theory canonically induces left- and right-stable algebraic dual adjunctions, and we establish a biequivalence between such algebras and left- (or right-)stable algebraic dual adjunctions, and also between saturated algebras and stable algebraic dual adjunctions. We also discuss examples of algebraic dual adjunctions, including dualization of internal modules, Pontryagin and Binz-Butzmann duality, dualization of internal affine spaces and convex spaces, dualization of semilattices, dualization of complete sup-lattices, and dualization of abelian groups with reference to a theorem of Ehrenfeucht and {\L}o\'s.