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We present a generalized theoretical framework for olfactory representation and plasticity, using the theory of smooth manifolds and sheaves to depict categorical odor learning via distributed neural computation. Beginning with the space of all possible inputs to the olfactory system, we develop a dynamic model for odor learning that culminates in a perceptual space in which categorical odor representations are hierarchically constructed through experience, exhibiting statistically appropriate consequential regions and clear relationships between the broader and narrower identities to which a given odor might be assigned. The model reflects both the sampling-based physical similarity relationships among odorants, as observed in physiological receptor response profiles, and the acquired, learning-dependent perceptual similarity relationships among odors that can be measured behaviorally, and defines the relationship between them. Individual training and experience generates correspondingly more sophisticated odor identification capabilities. Because these odor representations are constructed from experience and depend on local, distributed plasticity mechanisms, geometries that fix curvature are insufficient to describe the capabilities of the system. This generative framework also encompasses hypotheses explaining representational drift in postbulbar circuits and the context-dependent remapping of perceptual similarity relationships.