标题： 类别 显示英文标题

标题： Itegories

摘要： 一个it范畴是具有克利尼魔杖的限制范畴。 Cockett， Díaz-Boïls, Gallagher和Hrubeš简要介绍了克利尼魔杖以捕捉源于复杂性理论的限制范畴中的迭代。 本文的目的在于更详细地发展克利尼魔杖和it范畴的理论。 克利尼魔杖是一种二元运算符，它接受两个不相交的部分映射，一个自同构${X \to X}$和一个映射${X \to A}$，并产生一个部分映射$X \to A$。 后者被解释为迭代自同构直到它落在第二个映射定义域内。 在一个具有无限不相交并的环境中，总是存在一个规范的克利尼魔杖，通过实现这一直觉来定义。 迭代的标准范畴解释是通过余积上的迹运算完成的。 对于广泛的限制范畴，我们详细解释了拥有克利尼魔杖与迭代的标准解释等价。 这表明克利尼魔杖可以用来替代那些缺乏余积的限制范畴中的参数化迭代和迹。 进一步的证据体现在提供了一个矩阵构造，将一个it范畴嵌入到一个带有迹的广泛的限制范畴中。 我们也考虑了经典限制范畴中的克利尼魔杖，并展示了在这种情况下，一个克利尼魔杖完全由其自同构分量确定。 显示英文摘要

摘要： An itegory is a restriction category with a Kleene wand. Cockett, D\'iaz-Bo\"ils, Gallagher, and Hrube\v{s} briefly introduced Kleene wands to capture iteration in restriction categories arising from complexity theory. The purpose of this paper is to develop in more detail the theory of Kleene wands and itegories. A Kleene wand is a binary operator which takes in two disjoint partial maps, an endomorphism ${X \to X}$ and a map ${X \to A}$ and produces a partial map $X \to A$. This latter map is interpreted as iterating the endomorphism until it lands in the domain of definition of the second map. In a setting with infinite disjoint joins, there is always a canonical Kleene wand given by realizing this intuition. The standard categorical interpretation of iteration is via trace operators on coproducts. For extensive restriction categories, we explain in detail how having a Kleene wand is equivalent to this standard interpretation of iteration. This suggests that Kleene wands can be used to replace parametrized iteration and traces in restriction categories which lack coproducts. Further evidence of this is exhibited by providing a matrix construction which embeds an itegory into a traced extensive restriction category. We also consider Kleene wands in classical restriction categories and show how, in this case, a Kleene wand is completely determined by its endomorphism component.