In mathematics, specifically in category theory, an
-coalgebra is a structure defined according to a functor
, with specific properties as defined below.
For both algebras and coalgebras,[clarification needed] a functor is a convenient and general way of organizing a signature.
This has applications in computer science: examples of coalgebras include lazy evaluation, infinite data structures, such as streams, and also transition systems.
Just as the class of all algebras for a given signature and equational theory form a variety, so does the class of all
-coalgebras satisfying a given equational theory form a covariety, where the signature is given by
Let be an endofunctor on a category
-coalgebras for a given functor F constitute a category.
Consider the endofunctor
that sends a set to its disjoint union with the singleton set
is the so-called conatural numbers, consisting of the nonnegative integers and also infinity, and the function
is the terminal coalgebra of this endofunctor.
More generally, fix some set
is a finite or infinite stream over the alphabet
Applying the state-transition function to a state may yield two possible results: either an element of
together with the next state of the stream, or the element of the singleton set
as a separate "final state" indicating that there are no more values in the stream.
In many practical applications, the state-transition function of such a coalgebraic object may be of the form
, which readily factorizes into a collection of "selectors", "observers", "methods"
Special cases of practical interest include observers yielding attribute values, and mutator methods of the form
taking additional parameters and yielding states.
Let P be the power set construction on the category of sets, considered as a covariant functor.
The P-coalgebras are in bijective correspondence with sets with a binary relation.
Then coalgebras for the endofunctor P(A×(-)) are in bijective correspondence with labelled transition systems, and homomorphisms between coalgebras correspond to functional bisimulations between labelled transition systems.
In computer science, coalgebra has emerged as a convenient and suitably general way of specifying the behaviour of systems and data structures that are potentially infinite, for example classes in object-oriented programming, streams and transition systems.
While algebraic specification deals with functional behaviour, typically using inductive datatypes generated by constructors, coalgebraic specification is concerned with behaviour modelled by coinductive process types that are observable by selectors, much in the spirit of automata theory.
An important role is played here by final coalgebras, which are complete sets of possibly infinite behaviours, such as streams.
The natural logic to express properties of such systems is coalgebraic modal logic.
[citation needed]