Abstract
In this paper, we formalize precisely the sense in which the application of a cellular automaton to partial configurations is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis–Hedlund theorem of cellular automata from the extended topological point of view provided by category theory. These links also allow to relatively easily generalize concepts pioneered by cellular automata to arbitrary kinds of possibly evolving spaces. No prior knowledge of category theory is assumed for the most part.
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We chose the collection of objects and arrows to be sets. They are classes in usual definitions.
Note that one can only compose pairs of arrows where the first arrow starts where the second arrow ends, for instance \(g \circ f\) is defined for \(g : y \rightarrow z\) and \(f : x \rightarrow y\). This means that associativity, left neutrality and right neutrality only holds when they are defined.
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Fernandez, A., Maignan, L. & Spicher, A. Cellular automata and Kan extensions. Nat Comput 22, 493–507 (2023). https://doi.org/10.1007/s11047-022-09931-0
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DOI: https://doi.org/10.1007/s11047-022-09931-0