丰富代数理论的二重代数和交换子
Bifold algebras and commutants for enriched algebraic theories
摘要：
集合上代数结构的交换对已经被几个作者研究过，并且可以等价地描述为Lawvere理论的张量积的代数，或者更基本地描述为某些双函子，在这里我们称之为双折叠代数。Lawvere理论中较少研究的交换子概念首先由Wraith引入，并推广了泛代数中的中心化克隆概念。工作在一般的设置丰富的代数理论的一个系统的arities，我们研究的概念的相互作用的bifold代数和交换子。我们表明，交换子的概念产生通过一个普遍的建设在双边纤维化的bifold代数在各种理论。在此基础上，我们研究了与交换子相关的特殊双折叠代数类，引入了交换双折叠代数和平衡双折叠代数的概念。我们建立了几个aprections和等价这些类别的bifold代数和相关类别的代数在各种理论，包括交换，收缩交换，饱和和平衡代数。我们还调查和发展的例子交换bifold代数，包括使用庞特里亚金对偶和一个定理的Eschericht和Eschericht自反阿贝尔群的例子。一路上，我们开发了一个函处理的基本方面的bifold代数和交换，包括张量积的理论和等价的bifold代数和交换对代数。因为我们的工作相对于一个（可能是大的）系统的arities在一个封闭的范畴$\mathcal{V}$，我们的主要结果适用于任意$\mathcal{V}$ -单子上的一个完全的$\mathcal{V}$，丰富的理论Borceux和日，丰富的Lawvere理论的权力相对于一个经常基数，和其他概念的代数理论。
英文摘要：
Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category $\mathcal{V}$, our main results are applicable to arbitrary $\mathcal{V}$-monads on a finitely complete $\mathcal{V}$, the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory.
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