论文标题
从GS-konoidal到Oplax笛卡尔类别：构造和功能完整性
From Gs-monoidal to Oplax Cartesian Categories: Constructions and Functorial Completeness
论文作者
论文摘要
GS-Mononoidal类别的概念最初是在代数方法重写的代数方法的上下文中引入的，在过去的几十年中，在不同的绰号下已经浮出水面几次。可以将它们视为对称的单体类别，其箭头是广义关系，其结构足以谈论域和部分功能，但与笛卡尔主持人相比，结构较少。本文的目的是三倍。第一个目标是通过在箭头上的预订来丰富GS-Onoidality的原始定义，从而产生我们所谓的Oplax笛卡尔类别。其次，我们表明（富含预订的）GS - 单型类别自然是作为kleisli类别和跨度类别出现的，并且探索了由此产生的形式主义之间的关系。最后，我们提出了两个定理，一方面是关于yoneda嵌入的，另一方面是功能完整性，后者从oplax笛卡尔类别到$ \ mathbf {rel} $的洛杉矶力函数也引起完整的结果。
Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal category has surfaced a few times under different monikers in the last decades. They can be thought of as symmetric monoidal categories whose arrows are generalised relations, with enough structure to talk about domains and partial functions, but less structure than cartesian bicategories. The aim of this paper is threefold. The first goal is to extend the original definition of gs-monoidality by enriching it with a preorder on arrows, giving rise to what we call oplax cartesian categories. Second, we show that (preorder-enriched) gs-monoidal categories naturally arise both as Kleisli categories and as span categories, and the relation between the resulting formalisms is explored. Finally, we present two theorems concerning Yoneda embeddings on the one hand and functorial completeness on the other, the latter inducing a completeness result also for lax functors from oplax cartesian categories to $\mathbf{Rel}$.
