Applications of category theory often involve symmetric monoidal categories (SMCs), in which abstract processes or operations can be composed in series and parallel. However, in 2020 there remains a dearth of computational tools for working with SMCs. We present an "unbiased" approach to implementing symmetric monoidal categories, based on an operad of directed, acyclic wiring diagrams. Because the interchange law and other laws of a SMC hold identically in a wiring diagram, no rewrite rules are needed to compare diagrams. We discuss the mathematics of the operad of wiring diagrams, as well as its implementation in the software package Catlab.
翻译:类别理论的应用往往涉及对称单向类别(SMCs),在这些类别中,抽象过程或操作可以按序列和平行方式组成;然而,在2020年,仍然缺乏与SMCs合作的计算工具。我们提出了一个“无偏见的”方法来实施对称的单向类别,其依据是一组直线、周期性电线图。由于SMC的交换法和其他法律在电线图中相同,因此无需重写规则来比较图表。我们讨论了电线图剧的数学及其在软件包“Catlab”中的应用。