Itegories

This paper introduces Kleene wands, which capture guarded iteration in restriction categories. A Kleene wand is a binary operator which takes in two maps, an endomorphism $X \to X$ and a map ${X \to A}$, which are disjoint and and produces a map $X \to A$. This map is interpreted as iterating the endomorphism until it lands in the domain of definition of the second map, which plays the role of a guard. In a setting with infinite disjoint joins, there is always a canonical Kleene wand given by realizing this intuition. We call a restriction category with a Kleene wand an itegory. To provide further evidence that Kleene wands capture iteration, we explain how Kleene wands are deeply connected to trace operators on coproducts, which are already well-known of categorifying iteration. We show that for an extensive restriction category, to have a Kleene wand is equivalent to having a trace operator on the coproduct. This suggests, therefore, that Kleene wands can be used to replace parametrized iteration operators or trace operators in a setting without coproducts.