Fibers Over Standard Moves — Essential Non-Injectivity of the Forgetful Functor in Rule Categories

A game can be described by its possible moves, but these moves do not necessarily determine the rules that govern them.

We formalize this distinction using a category R of rule systems and a category T of rooted directed graphs. The forgetful functor U_{p₀} : R → T preserves the move structure while forgetting the values assigned to terminal positions. This allows us to compare different rule systems over the same move structure.

Using the example of the Scholar’s Mate graph, we show that U_{p₀} admits no left inverse up to isomorphism: distinct rules can overlap over the same structure of play.