Universal coalgebra

Summary

Universal coalgebra is the general study of coalgebras for an endofunctor of the category of sets. Such a coalgebra is a set of states $X$ with a map $X \to F(X)$. The map tells what can be observed of a state and which states come next. It is the dual of universal algebra: algebras build elements from smaller ones, coalgebras unfold states into their behaviour. Streams, automata, labelled transition systems and many other state-based systems are coalgebras, and the theory treats their behaviour and their bisimulations in one framework. The general notion over any category is the F-coalgebra.