esquisses catégories adt spécifications

Category theory is more and more used in studying abstract data types. Since long time, some authors used the notion of sketch to study the syntax and semantics of mathematical structures. This notion, introduced and developed by Ehresmann, is more powerful than the signature one, since it uses limits in categorical sense. Indeed, the signature approach uses nothing but products, while the notion of sketch allows not only to specify algebraic structures but also non algebraic ones such as fields.

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categories adt

We show how to define fold operators for abstract data types. The main idea is to represent an ADT by a bialgebra, that is, an algebra/coalgebra pair with a common carrier. A program operating on an ADT is given by a mapping to another ADT. Such a mapping, called metamorphism, is basically a composition of the algebra of the second with the coalgebra of the first ADT. We investigate some properties of metamorphisms, and we show that the metamorphic programming style offers far-reaching opportunities for program optimization that cover and even extend those known for algebraic data types.

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