Coarse graining is defined in terms of a commutative diagram. Necessary and sufficient conditions are given in the continuously differentiable case. The theory is applied to linear coarse grainings arising from partitioning the population space of a simple Genetic Algorithm (GA). Cases considered include proportional selection, binary tournament selection, and mutation. A nonlinear coarse graining for ranking selection is also presented. Within the context of GAs, the primary contribution made is the introduction and illustration of a technique by which the possibility for coarse grainings may be analyzed. A secondary contribution is that a number of new coarse graining results are obtained.
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