Mongruences and Cofree Coalgebras

A coalgebra is introduced here as a model of a certain signature consisting of a type X with various \destructor" function symbols, satisfying certain equations. These destructor function symbols are like methods and attributes in object-oriented programming: they provide access to the type (or state) X. We show that the category of such coalgebras and structure preserving functions is comonadic over sets. Therefore we introduce the notion of a `mongruence' (predicate) on a coalgebra. It plays the dual role of a congrence (relation) on an algebra. An algebra is a set together with a number of operations on this set which tell how to form (derived) elements in this set, possibly satisfying some equations. A typical example is a monoid, given by a set M with operations 1 ! M, M M ! M. Here 1 = f;g is a singleton set. In mathematics one usually considers only single-typed algebras, but in computer science one more naturally uses many-typed algebras like 1 ! list(A), A list(A) ! list(A). Here we are not primarily interested in algebras, but in coalgebras. These consist of a set together with operations \going out" of the set; they tell how to \deconstruct" elements of the set. These kind of structures naturally arise within object-oriented programming|as explained in 16, 10], but see also 13], or 5] (where the \destructors" are called \observers"). A possible (set-theoretic) semantics for objects is to see them as pairs hx 2 X; c: X ! T(X)i where c is a coalgebra of a functor T: Sets ! Sets interpreting the methods, and x is an element of the carrier X of the coalgebra c. Usually c will be a tuple c = hc 1 ; : : :; c n i of maps. Message passing then means application of the appropriate component of c to x 2 X. This picture is very elementary but clear enough to serve as our motivation. Here we concentrate on equations in such coalgebras. The main conceptual novelty that we introduce is the notion of what we call a mongruence (predicate). A mongruence plays the same role in coalgebra that a congruence plays in algebra: it is a predicate which is closed under the coalgebra operations, just like a congruence is a relation which is closed under the algebra operations. We use these mongruences in particular to construct cofree coalgebras satisfying certain equations. In Mac Lane's …