Games as recursive coalgebras: A categorical view on the Nim-sum

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.