  The traditional mathematical model for an impartial combinatorial game is
defined recursively as a set of the options of the game, where the options are
games themselves. We propose a model called gamegraph, together with its
generalization rulegraph, based on the natural description of a game as a
digraph where the vertices are positions and the arrows represent possible
moves. Such digraphs form a category where the morphisms are option preserving
maps. We study several versions of this category. Our development includes
congruence relations, quotients, and isomorphism theorems and is analogous to
the corresponding notions in universal algebra. The quotient by the maximum
congruence relation produces an object that is essentially equivalent to the
traditional model. After the development of the general theory, we count the
number of non-isomorphic gamegraphs and rulegraphs by formal birthday and the
number of positions.
