When game comparison becomes play: Absolutely Categorical Game Theory

Abstract

Absolute Universes of combinatorial games, as defined in a recent paper by the same authors, include many standard short normal- misère- and scoring-play monoids. In this note we show that the class is categorical, by extending Joyal's construction of arrows in normal-play games. Given $G$ and $H$ in an Absolute Universe $U$, we study instead the Left Provisonal Game $[G,H]$, which is a normal-play game, independently of the particular Absolute Universe, and find that $G⟶H$ (implying $G\succcurlyeq H$) corresponds to the set of winning strategies for Left playing second in $[G,H]$. By this we define the category $\mathbf{LNP}(\mathbf{U})$.

14 pages. arXiv admin note: text overlap with arXiv:1606.01975