Itegory（迭代范畴）
Itegories
AI总结：
本文发展了Kleene wand与itegory的理论，阐释了广延限制范畴中Kleene wand与标准迭代解释的等价性，证明其可替代缺余积限制范畴的参数化迭代与迹，并研究了经典限制范畴中Kleene wand的决定因素。
摘要：
itegory是配备Kleene wand（克林枝）的限制范畴。Cockett、Díaz-Boïls、Gallagher与Hrubeš曾简要引入Kleene wand，用于刻画复杂性理论中产生的限制范畴里的迭代。本文的目的是更详尽地发展Kleene wand与itegory的理论。Kleene wand是一种二元算子，它接收两个不相交的偏映射，一个类型为${X \to X}$，另一个类型为${X \to A}$，并生成一个类型为$X \to A$的偏映射。后一个映射被解释为迭代自同态，直到它落入第二个映射的定义域。在存在无限不相交并的设定中，始终存在一个典范的Kleene wand来实现这一直觉。迭代的标准范畴解释是通过余积上的迹算子实现的。对于广延限制范畴，我们详细阐释了拥有Kleene wand等价于这种标准的迭代解释。这表明，在缺少余积的限制范畴中，Kleene wand可用于替代参数化迭代和迹。我们通过提供一种将itegory嵌入到带迹的广延限制范畴中的矩阵构造，进一步证明了这一点。我们还研究了经典限制范畴中的Kleene wand，并说明在这种情况下，Kleene wand完全由其自同态分量决定。
英文摘要：
An itegory is a restriction category with a Kleene wand. Cockett, D{\'ı}az-Bo{\"ı}ls, Gallagher, and Hrube{š} 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, one of type ${X \to X}$ and the other of type ${X \to A}$, 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 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.
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