Regular logic can be regarded as the internal language of regular categories, but the logic itself is generally not given a categorical treatment. In this paper, we understand the syntax and proof rules of regular logic in terms of the free regular category FRg(T) on a set T. From this point of view, regular theories are certain monoidal 2-functors from a suitable 2-category of contexts -- the 2-category of relations in FRg(T) -- to that of posets. Such functors assign to each context the set of formulas in that context, ordered by entailment. We refer to such a 2-functor as a regular calculus because it naturally gives rise to a graphical string diagram calculus in the spirit of Joyal and Street. We shall show that every natural category has an associated regular calculus, and conversely from every regular calculus one can construct a regular category.

中文翻译：

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常规逻辑的字符串图（扩展摘要）

正则逻辑可以看成是正则范畴的内在语言，但一般不会对逻辑本身进行范畴化处理。在本文中，我们根据集合 T 上的自由正则范畴 FRg(T) 来理解正则逻辑的语法和证明规则。 从这个角度来看，正则理论是来自一个合适的 2-范畴的某些幺半群 2-函子上下文——FRg(T) 中的 2 类关系——到poset 的关系。这样的函子为每个上下文分配该上下文中的一组公式，按蕴涵排序。我们将这样的 2-函子称为正则微积分，因为它本着 Joyal 和 Street 的精神自然而然地产生了图形字符串图微积分。我们将证明每个自然范畴都有一个相关的正则演算，