AlChemy models chemistry by treating molecules as lambda calculus terms and reactions as reductions. Two terms collide; one applies to the other; the result is a new term — a new molecule. The system generates emergent complexity because reduction can produce terms more complex than either input. But AlChemy's algebraic structure (what reactions are possible) and its dynamics (which reactions happen) were connected informally.
Bournez et al. formalize the connection as a functor. The algebraic model — the space of terms and their reduction rules — lives in one category. The dynamical model — the time evolution of populations of terms — lives in another. The functor maps algebraic structure to dynamical behavior, ensuring that every algebraic property has a corresponding dynamical consequence and vice versa.
The generalization matters because it separates what can react from what does react. In the algebraic category, every valid reduction is present. In the dynamical category, concentrations, collision rates, and selection pressures determine which reductions actually occur. The functor preserves the structure while allowing the dynamics to select from it. This is the same separation that physical chemistry enforces: thermodynamics determines what reactions are possible; kinetics determines which ones happen.
The through-claim is about the formalization gap in artificial life. AlChemy worked — it produced self-organizing chemical networks, autocatalytic sets, parasitism, and evolution. But without a formal bridge between algebra and dynamics, it was unclear which emergent properties were consequences of the algebraic structure and which were artifacts of the particular dynamical implementation. The functor makes this distinction precise.
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