For a category and endofunctor , an algebra of is an object in and a map . ( is called the carrier of the algebra)
A morphism between two algebras and of is a morphism in such that the following square commutes:
Composition of such morphisms of algebras is given by composition of the underlying morphisms in .
The dual concept is a coalgebra for an endofunctor. Both algebras and coalgebras for endofunctors on are special cases of algebras for C-C bimodules.
If the endofunctor has the structure of a monad, then an algebra (aka module) over that monad is an algebra for that satisfies an additional associativity property. Morphisms between algebras for monads are simply morphisms between algebras for the underlying functors. For more details see algebra over a monad.
An initial object in the category of algebras for is an initial algebra of .