Games as recursive coalgebras: A categorical view on the Nim-sum
Ryuya Hora
TL;DR
This work provides a categorical reinterpretation of impartial games by modeling them as recursive $\mathcal{P}_{\mathrm{fin}}$-coalgebras, unifying Grundy numbers, Conway addition, and nim-sum under a single coalgebraic framework. The category $\mathbf{Game}$ is shown to be equivalent to the category of recursive $\mathcal{P}_{\mathrm{fin}}$-coalgebras, with game values realized as hylo morphisms into $\mathcal{P}_{\mathrm{fin}}$-algebras; the terminal object $\mathbb{H}$ consists of hereditarily finite sets, tied to Adámek's fixed-point construction. A central contribution is the Bouton monoid, a universal, minimal monoid structure that decomposes a game into subgames and synthesizes game values under a chosen monoidal sum; this framework recovers the classical Nim-sum via the Conway addition and outcome, and yields concrete examples for selective and conjunctive sums. The paper also develops a robust set of categorical properties for $\mathbf{Game}$ (locally finitely presentable, comonadic over Set, with a subobject classifier) and lays out seven open questions, including differential structures, partisan extensions, and monoidal-classification problems. Overall, the work provides a principled, scalable foundation for generalizing nim-like rules and analyzing complex game decompositions using category-theoretic methods, with potential implications for broader combinatorial game theory and logic.
Abstract
In 1901, Bouton proved that a winning strategy of the game of Nim is given by the bitwise XOR, called the nim-sum. But, why does such a weird binary operation work? Led by this question, this paper introduces a categorical reinterpretation of combinatorial games and the nim-sum. The main categorical gadget used here is recursive coalgebras, which allow us to redefine games as ``graphs on which we can conduct recursive calculation'' in a concise and precise way. For game-theorists, we provide a systematic framework to decompose an impartial game into simpler games and synthesize the quantities on them, which generalizes the nim-sum rule for the Conway addition. To read the first half of this paper, the categorical preliminaries are limited to the definitions of categories and functors. For category theorists, this paper offers a nicely behaved category of games $\mathbf{Game}$, which is a locally finitely presentable symmetric monoidal closed category comonadic over $\mathbf{Set}$ admitting a subobject classifier! As this paper has several ways to be developed, we list seven open questions in the final section.