States of the brain represent states of the world. But at least some of the mind–brain's internal representations, such as a sensation of heat or a sensation of red, do not resemble the external realities that they represent: mean kinetic energy (temperature) or electromagnetic reflectance (colour). The historical response has been to distinguish between objectively real properties, such as shape and motion, and subjective properties, such as heat and colour. However, this approach leads to trouble. A challenge for cognitive neurobiology is to characterize, in general terms, the relationship between brain models and the world. We propose that brains develop high-dimensional maps, the internal distance relationships of which correspond to the similarity relationships that constitute the categorical structure of the world.