An introduction to (co)algebra and (co)induction

Abstract

Algebra is a well-established part of mathematics, dealing with sets with operations satisfying certain properties, like groups, rings , vector spaces, etcetera. Its results are essential throughout mathematics and other sciences. Universal algebra is a part of algebra in which algebraic structures are studied at a high level of abstraction and in which general notions like homomorphism, subalge-bra, congruence are studied in themselves, see e.g. [Coh81, MT92, Wec92]. A further step up the abstraction ladder is taken when one studies algebra with the notions and tools from category theory. This approach leads to a particularly concise notion of what is an algebra (for a functor or for a monad), see for example [Man74]. The conceptual world that we are about to enter owes much to this categorical view, but it also takes inspiration from universal algebra, see e.g. [Rut00]. In general terms, a program in some programming language manipulates data. During the development of computer science over the past few decades it became clear that an abstract description of these data is desirable, for example to ensure that one's program does not depend on the particular representation of the data on which it operates. Also, such abstractness facilitates correctness proofs. This desire led to the use of algebraic methods in computer science, in a branch called algebraic specification or abstract data type theory. The object of study are data types in themselves, using notions and techniques which are familiar from algebra. The data types used by computer scientists are often generated from a given collection of (constructor) operations. The same applies in fact to programs, which themselves can be viewed as data too. It is for this reason that " initiality " of algebras plays such an important role in computer science (as first clearly emphasised in [GTW78]). See for example [EM85, Wir90, Wec92] for more information.