The classic problem of stimulus-response (S-R) compatibility (SRC) is addressed. A cognitive model is proposed that views the stimulus and response sets in S-R ensembles as categories with dimensions that may or may not overlap. If they do overlap, the task may be compatible or incompatible, depending on the assigned S-R mapping. If they do not overlap, the task is noncompatible regardless of the assigned mapping. The overlapping dimensions may be relevant or not. The model provides a systematic account of SRC effects, a taxonomy of simple performance tasks that were hitherto thought to be unrelated, and suggestive parallels between these tasks and the experimental paradigms that have traditionally been used to study attentional, controlled, and automatic processes.