We define varieties of algebras for an arbitrary endofunctor on a cocomplete category using pairs of natural transformations. This approach is proved to be equivalent to one of equational classes defined by equation arrows. Free algebras in the varieties are investigated and their existence is proved under the assumptions of accessibility.
The purpose of this paper is two fold: we study the behaviour of the forgetful functor from -modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras.Let denote the forgetful functor from -modules to graded vector spaces. Left modules over an operad are treated as -algebras in the category of -modules. We generalize the results obtained...