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The costructure-cosemantics adjunction for comodels for computational effects
arXiv - CS - Logic in Computer Science Pub Date : 2020-11-30 , DOI: arxiv-2011.14520
Richard Garner

It is well established that equational algebraic theories, and the monads they generate, can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory T -- i.e., models in the opposite category Set^op -- provide a suitable environment for evaluating the computational effects encoded by T. As already noted by Power and Shkaravska, taking comodels yields a functor from accessible monads to accessible comonads on Set. In this paper, we show that this functor is part of an adjunction -- the "costructure-cosemantics adjunction" of the title -- and undertake a thorough investigation of its properties. We show that, on the one hand, the cosemantics functor takes its image in what we term the presheaf comonads induced by small categories; and that, on the other, costructure takes its image in the presheaf monads induced by small categories. In particular, the cosemantics comonad of an accessible monad will be induced by an explicitly-described category called its behaviour category that encodes the static and dynamic properties of the comodels. Similarly, the costructure monad of an accessible comonad will be induced by a behaviour category encoding static and dynamic properties of the comonad coalgebras. We tie these results together by showing that the costructure-cosemantics adjunction is idempotent, with fixpoints to either side given precisely by the presheaf monads and comonads. Along the way, we illustrate the value of our results with numerous examples drawn from computation and mathematics.

中文翻译:

用于计算效果的协模型的协结构-句法附加

众所周知,方程式代数理论及其生成的单子可用于编码计算效果。Power和Shkaravska的一个重要见解是,代数理论T的协模型(即Set ^ op相反类别的模型)为评估T编码的计算效果提供了合适的环境。Power和Shkaravska已经指出,采取协同模型可产生从Set上可访问的monad到可访问的comonads的函子。在本文中,我们证明了该函子是附加项的一部分-标题的“共结构-语法附加项”,并对它的性质进行了彻底的研究。我们表明,一方面,同感函子将其图像描述为小类别引起的前鞘同名。另一方面,共结构在由小类引起的前叶单核细胞中表现出来。特别是,可访问单子的同义语义将由一个明确描述的类别(称为行为类别)来诱发,该类别对协模型的静态和动态属性进行编码。类似地,可访问共面体的共结构单峰将由编码共面体代数的静态和动态特性的行为类别引起。我们通过显示共结构-句法辅助词是等幂的,将这些结果结合在一起,并通过前束单子和共母精确地给定任一侧的固定点。在此过程中,我们使用大量的计算和数学示例来说明结果的价值。可访问单子的共语义将由一个明确描述的类别(称为行为类别)来诱发,该类别对协模型的静态和动态属性进行编码。类似地,可访问共面体的共结构单峰将由编码共面体代数的静态和动态特性的行为类别引起。我们通过显示共结构-句法辅助词是等幂的,将这些结果结合在一起,并通过前束单子和共母精确地给定任一侧的固定点。在此过程中,我们使用大量的计算和数学示例来说明结果的价值。可访问单子的共语义将由一个明确描述的类别(称为行为类别)来诱发,该类别对协模型的静态和动态属性进行编码。类似地,可访问共面体的共结构单峰将由编码共面体代数的静态和动态特性的行为类别引起。我们通过显示共结构-句法辅助词是等幂的,将这些结果结合在一起,并通过前束单子和共母精确地给定任一侧的固定点。在此过程中,我们使用大量的计算和数学示例来说明结果的价值。可访问共面体的共结构单峰将由编码共面体代数的静态和动态特性的行为类别引起。我们通过显示共结构-句法辅助词是等幂的,将这些结果结合在一起,并通过前束单子和共母精确地给定任一侧的固定点。在此过程中,我们使用大量的计算和数学示例来说明结果的价值。可访问共面体的共结构单峰将由编码共面体代数的静态和动态特性的行为类别引起。我们通过显示共结构-句法辅助词是等幂的,将这些结果结合在一起,并通过前束单子和共母精确地给定任一侧的固定点。在此过程中,我们使用大量的计算和数学示例来说明结果的价值。
更新日期:2020-12-01
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