Itegories

2025-04-03

AI Summary

Formalizing guarded iteration is challenging in restriction categories lacking coproducts, as standard approaches rely on coproduct-based trace operators or parameterized iteration. Method: We introduce the *Kleene wand*, a novel binary operator that characterizes the controlled repeated application of a morphism (X o X) to a morphism (X o A) with disjoint domains, yielding (X o A). Contribution/Results: This constitutes the first axiomatization of guarded iteration in restriction categories without coproducts. Crucially, we establish an equivalence between the Kleene wand and trace operators on coproducts: in extensive restriction categories, a Kleene wand exists if and only if a trace operator does. Consequently, the Kleene wand serves as a coproduct-free alternative to parameterized iteration or trace-based iteration, providing a new foundational axiomatization for iteration. This advances the interface between Kleene algebra semantics and restriction category theory, broadening their applicability to settings where coproducts are unavailable or unnatural.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

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 $X o X$ and a map ${X o A}$, which are disjoint and and produces a map $X o A$. 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.

Problem

• Captures guarded iteration in restriction categories
• Connects Kleene wands to trace operators on coproducts
• Replaces parametrized iteration in settings without coproducts

Innovation

• Kleene wands enable guarded iteration
• Connects wands to trace operators
• Replaces iteration operators without coproducts