Algebras and modules
Model category presentations
Geometry on formal duals of algebras
An algebra over an endofunctor is like an algebra over a monad, but without a notion of associativity (which would not make sense).
For a category and endofunctor , an algebra (or module) of is an object in and a morphism . ( is called the carrier of the algebra)
A homomorphism 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 . This yields the category of -algebras, which comes with a forgetful functor to .
Relation to algebras over a monad
To a category theorist, algebras over a monad may be more familiar than algebras over just an endofunctor. In fact, when and are well-behaved, then algebras over an endofunctor are equivalent to algebras over a certain monad, the algebraically-free monad generated by .
This is analogous to the relationship between an action of a monoid and a binary function (an action of a set): such a function is the same thing as an action of the free monoid on .
Returning to the endofunctor case, the general statement is:
Actually, this proposition is merely a definition of the term “algebraically-free monad”. If has an algebraically-free monad, denoted say , then in particular the forgetful functor has a left adjoint, and is the monad on generated by this adjunction. Conversely, if such a left adjoint exists, then the monad it generates is algebracially-free on ; for the straightforward proof, see for instance (Maciej).
Algebraically-free monads exist in particular when is a locally presentable category and is an accessible functor; see transfinite construction of free algebras.
A textbook account of the basic theory is in chapter 10 of
The relation to free monads is discussed in