Piaget introduced a category-theoretic formulation of cognitive development in the late 1960s and extended it during the 1970s. The new theory, which has received surprisingly little attention from cognitive developmentalists, is interpreted in this article as an organizing theme of Piaget’s investigations into such diverse topics as functions, correspondences, commutability, equilibration, reflective abstraction, and the opening of possibilities. The concept of morphism, which is central to the category-theoretic formalism, is discussed and examples of tasks useful for studying this concept are described. A number of implications of the new model and possible future directions are then explored. It is concluded that the new formulation offers the potential for an integrated model of cognitive process and cognitive structure, and is therefore a significant contribution to constructivist theory.