Categorical products of cellular automata

TL;DR

The paper addresses how to formulate and verify product-like constructions in categories of cellular automata, distinguishing CA over a fixed group from generalized CA between different groups. It shows that CA over a fixed group $G$ has ordinary categorical products realized by $$(A_1 \times A_2)^G$$ with componentwise projections, while generalized CA have a weak product realized by $$(A_1 \times A_2)^{G_1 * G_2}$$, reflecting the free product structure. It provides explicit constructions for product morphisms and weak product morphisms and discusses when the weak product fails to be a product due to lack of unique homomorphism property. The results establish a principled categorical framework for combining cellular automata across identical or different groups, enabling natural product-like operations and offering a structural lens for CA design and analysis.

Abstract

We study two categories of cellular automata. First, for any group $G$, we consider the category $\mathcal{CA}(G)$ whose objects are configuration spaces of the form $A^G$, where $A$ is a set, and whose morphisms are cellular automata of the form $τ: A_1^G \to A_2^G$. We prove that the categorical product of two configuration spaces $A_1^G$ and $A_2^G$ in $\mathcal{CA}(G)$ is the configuration space $(A_1 \times A_2)^G$. Then, we consider the category of generalized cellular automata $\mathcal{GCA}$, whose objects are configuration spaces of the form $A^G$, where $A$ is a set and $G$ is a group, and whose morphisms are $φ$-cellular automata of the form $\mathcal{T} : A_1^{G_1} \to A_2^{G_2}$, where $φ: G_2 \to G_1$ is a group homomorphism. We prove that a categorical weak product of two configuration spaces $A_1^{G_1}$ and $A_2^{G_2}$ in $\mathcal{GCA}$ is the configuration space $(A_1 \times A_2)^{G_1 \ast G_2}$, where $G_1 \ast G_2$ is the free product of $G_1$ and $G_2$. The previous results allow us to naturally define the product of two cellular automata in $\mathcal{CA}(G)$ and the weak product of two generalized cellular automata in $\mathcal{GCA}$.

Figures (6)

• Figure 1: Definition of a product to two objects in a category.
• Figure 2: Product to two morphisms in a category.
• Figure 3: Definition of the coproduct to two objects in a category.
• Figure 4: Product of two configuration spaces in $\mathcal{CA}(G)$.
• Figure 5: Proof of the existence of a weak product in $\mathcal{GCA}$.

Theorems & Definitions (19)

• Example 1
• Definition 1: product
• Example 2
• Theorem 1
• proof
• Example 3
• Definition 2: coproduct
• Example 4
• Definition 3
• Remark 1