F-algebra

Summary

For an endofunctor $F$ of a category $\mathcal{C}$, an $F$-algebra is an object $A$ with a morphism $F A \to A$. For a suitable functor on sets, $F$-algebras are exactly the algebras of a given signature, so the notion generalizes universal algebra without equations. Inductive data types are initial F-algebras, and the dual notion, F-coalgebra, models state-based systems. Unlike an algebra over a monad, an $F$-algebra need not satisfy any unit or associativity law.