A Universal Construction for (Co)Relations
A Universal Construction for (Co)Relations
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（合作）关系的普遍构建
DOI：
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
F. Zanasi
中科院分区：
文献类型：
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作者：
Brendan Fong;F. Zanasi
弦图的演算越来越多地用于表示各种电路的语法和代数结构，包括信号流图、电路和量子过程。在许多这样的方法中，图表的语义解释是根据某种关系或相关关系（广义等价关系）给出的。在本文中，我们展示了关系和相关的语义范畴如何被表征为更简单范畴的边界。这种模块化的观点很重要，因为它简化了给出字符串图语义等价的完整公理的任务。此外，我们的一般结果统一了文献中独立发现的各种定理，包括线性相关关系（与电路语义相关），部分等价关系和线性子空间（信号流图和无相ZX演算的语义）的情况。
Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature, including the cases of linear corelations (relevant for the semantics of electrical circuits), of partial equivalence relations and of linear subspaces (semantics of signal flow graphs and of the phase-free ZX calculus).