丰富代数范畴间的对偶伴随
Dual adjunctions between enriched algebraic categories
浏览论文内容
中文总结 AI 辅助
该研究在丰富代数理论框架下，将 Freyd 定理推广至代数对偶伴随，建立其与双折叠代数特殊类的双等价，证明丰富代数理论的代数可诱导稳定代数对偶伴随并讨论相关实例。
中文摘要 AI 辅助
在具有一组 arity 的丰富代数理论框架下，我们研究丰富代数范畴间对偶伴随（称为代数对偶伴随）的若干方面。首先，我们将 Freyd 关于反变代数值右伴随函子的定理推广至该框架。其次，我们在作者前期工作定义的意义下，建立代数对偶伴随的 2-范畴与双折叠代数的局部离散 2-范畴之间的双等价。第三，我们定义代数对偶伴随的特殊类，称为（左和右）稳定代数对偶伴随，其中某些自由对象是自反的，我们建立这些特殊类与作者前期工作中基于交换子定义的双折叠代数特殊类之间的双等价。第四，我们证明每个丰富代数理论的代数可典范诱导左和右稳定代数对偶伴随，建立此类代数与左（或右）稳定代数对偶伴随之间的双等价，以及饱和代数与稳定代数对偶伴随之间的双等价。我们还讨论代数对偶伴随的例子，包括内部模的对偶、Pontryagin 与 Binz-Butzmann 对偶、内部仿射空间与凸空间的对偶、半格的对偶、完全上格的对偶，以及结合 Ehrenfeucht-Łoś 定理的阿贝尔群的对偶。
英文摘要
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 Łoś.
