A categorical framework for cellular automata

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.

💡 Research Summary

The paper presents a fully categorical formulation of cellular automata that works in any category 𝒞 equipped with finite products. The authors replace the classical alphabet set with an arbitrary object A of 𝒞 and the configuration space with the product A^G = ∏_{g∈G} A, where G is a group serving as the universe. By interpreting the right‑translation map R_g : G → G as a permutation, they define a pull‑back functor on products, yielding an automorphism φ_g = R_g^* of A^G. This automorphism reproduces the usual shift action of G on configurations, but now lives entirely inside the categorical framework.

A 𝒞‑cellular automaton is defined as a morphism τ : A^G → B^G that is locally determined: there exists a finite “memory set” S ⊂ G and a local rule μ : A^S → B such that for any configuration x ∈ A^G and any group element g, the new state at position g is τ(x)(g) = μ( (φ_{g^{-1}}(x))|_S ). This mirrors the classical definition while abstracting away from set‑theoretic notions.

The central theoretical contribution is a categorical version of the Curtis‑Hedlund‑Lyndon (CHL) theorem. The authors prove that a morphism τ : A^G → B^G is a 𝒞‑cellular automaton if and only if it satisfies two purely categorical conditions: (i) G‑equivariance, i.e., φ_g ∘ τ = τ ∘ φ_g for all g ∈ G, and (ii) uniform continuity with respect to the product uniformity on A^G. This result shows that the essence of CHL is categorical rather than topological, and it holds for any concrete or abstract category possessing products.

Next, the paper shows that the collection of 𝒞‑cellular automata over a fixed group G forms a subcategory CA_𝒞(G) of 𝒞. Moreover, CA_𝒞(G) is closed under finite products: the product of (A^G → B^G) and (C^G → D^G) is naturally identified with ((A×C)^G → (B×D)^G). This closure mirrors the classical fact that the product of two cellular automata is again a cellular automaton, but now it is proved using only universal properties of products.

The authors then generalize the construction to two possibly different groups G and H linked by a group homomorphism φ : H → G. A generalized 𝒞‑cellular automaton is a morphism τ : A^G → B^H that is locally defined via the same finite memory set, but the shift action on the target side is pulled back along φ. This extends the recent Set‑based work on “cellular automata between different universes” to an arbitrary category 𝒞.

Finally, they prove that the category GCA_𝒞 of generalized 𝒞‑cellular automata possesses a finite weak product. Given objects A^G and B^H, their weak product is (A×B)^{G∗H}, where G∗H denotes the free product of groups. The construction uses the product A×B in 𝒞 together with the free product group to index the configuration space. The weak product satisfies the universal property up to non‑uniqueness of the mediating morphism, which is sufficient for the categorical development pursued.

Overall, the paper makes several notable contributions:

1. Generalization – It lifts the definition of cellular automata from the concrete category Set to any category with products, encompassing both concrete (Grp, Top, Vect_K, Poset) and abstract (Poset(P), Rel) examples.
2. Categorical CHL theorem – It provides a clean, purely categorical proof that G‑equivariance plus uniform continuity characterizes cellular automata, emphasizing that the theorem’s core is categorical.
3. Closure properties – It shows that the resulting categories are closed under finite products and, in the generalized setting, under weak products involving free products of groups.
4. Unified framework – By avoiding any reliance on underlying set structures, the framework can be applied to settings where configurations are not sets but more exotic objects (e.g., relations, lattices).

The paper’s strengths lie in its conceptual clarity and the elegance of the categorical proofs. However, some limitations are evident: the exposition of concrete examples in non‑concrete categories is brief, leaving readers without a clear sense of how the theory behaves in Rel or Poset(P). Moreover, the treatment of uniform structures is abstract; a more explicit construction of the product uniformity in categories lacking a canonical topology would strengthen the applicability. Finally, computational aspects (e.g., how to simulate such automata) are not addressed, which may limit immediate impact on applied cellular automata research.

In conclusion, this work substantially broadens the theoretical foundations of cellular automata, positioning them within the language of category theory and opening avenues for further exploration in both pure mathematics (e.g., connections with topos theory, monoidal categories) and interdisciplinary applications where the underlying “states” are naturally categorical objects.