A categorical framework for cellular automata

TL;DR

This paper introduces a categorical framework for cellular automata, generalizing classical definitions using category theory, and proves foundational theorems within this abstract setting.

Contribution

It extends cellular automata theory to arbitrary categories with products, providing a unified, purely categorical approach and new proofs of key results.

Findings

Defines $ ext{C}$-cellular automata as morphisms in a category

Shows $ ext{C}$-cellular automata form a subcategory closed under finite products

Establishes a categorical version of the Curtis-Hedlund-Lyndon theorem

Abstract

This paper proposes a generalized framework for cellular automata using the language of category theory, extending the classical definition beyond set-theoretic constraints. For an arbitrary category $\mathscr{C}$ with products, we define $\mathscr{C}$-cellular automata as morphisms $\tau : A^G \to B^G$ in $\mathscr{C}$, where the alphabets $A$ and $B$ are objects in $\mathscr{C}$ and the universe is a group $G$. We show that $\mathscr{C}$-cellular automata form a subcategory of $\mathscr{C}$ closed under finite products, and that they satisfy a categorical version of the Curtis-Hedlund-Lyndon theorem. For two arbitrary group universes $G$ and $H$, we extend our theory to define generalized $\mathscr{C}$-cellular automata as morphisms $\tau : A^G \to B^H$ constructed via a group homomorphism $\phi : H \to G$. Finally, we prove that generalized $\mathscr{C}$-cellular automata form a subcategory of $\mathscr{C}$ with a finite weak product involving the free product of the underlying group universes. This framework unifies existing concepts and provides purely categorical proofs of foundational results in the theory of cellular automata.