  A sender sells an object of unknown quality to a receiver who pays his
expected value for it. Sender and receiver might hold different priors over
quality. The sender commits to a monotonic categorization of quality. We
characterize the sender's optimal monotonic categorization. Using our
characterization, we study the optimality of full pooling or full separation,
the alternation of pooling and separation, and make precise a sense in which
pooling is dominant relative to separation. We discuss applications, extensions
and generalizations, among them the design of a grading scheme by a
profit-maximizing school which seeks to signal student qualities and
simultaneously incentivize students to learn. Such incentive constraints force
monotonicity, and can also be embedded as a distortion of the school's prior
over student qualities, generating a categorization problem with distinct
sender and receiver priors.
