A categorical framework for cellular automata
A. Castillo-Ramirez, A. Vazquez-Aceves, A. Zaldivar-Corichi
TL;DR
The paper generalizes cellular automata to arbitrary categories with finite products by replacing the alphabet with a $\mathscr C$-object and working with configuration objects $A^{G}$. It defines $\mathscr C$-cellular automata as morphisms $\tau:A^{G}\to B^{G}$ with a finite neighborhood and local defining morphism, and proves a categorical Curtis–Hedlund–Lyndon theorem equating $\tau$ with $G$-equivariance and uniformity. It then extends to generalized $\mathscr C$-cellular automata between $A^{G}$ and $B^{H}$ via group homomorphisms $\phi:H\to G$, showing factorization through $\tau_G$ and the existence of weak products via the free product $G*H$, yielding a unified categorical framework. The approach yields purely categorical proofs of foundational results and applies across both concrete and abstract categories, broadening the scope of cellular automata theory.
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 $τ: A^G \to B^G$ 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 $τ: A^G \to B^H$ constructed via a group homomorphism $φ: H \to G$. 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.