Polynomial functor

In algebra, a polynomial functor is an endofunctor on the category formula_1 of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers formula_2 and the exterior powers formula_3 are polynomial functors from formula_1 to formula_1; these two are also Schur functors.
The notion appears in representation theory as well as category theory (the calculus of functors). In particular, the category of homogeneous polynomial functors of degree "n" is equivalent to the category of finite-dimensional representations of the symmetric group formula_6 over a field of characteristic zero.
Definition.
Let "k" be a field of characteristic zero and formula_1 the category of finite-dimensional "k"-vector spaces and "k"-linear maps. Then an endofunctor formula_8 is a "polynomial functor" if the following equivalent conditions hold:
A polynomial functor is said to be "homogeneous of degree "n if for any linear maps formula_16 in formula_1 with common domain and codomain, the vector-valued polynomial formula_18 is homogeneous of degree "n".
Variants.
If “finite vector spaces” is replaced by “finite sets”, one gets the notion of combinatorial species (to be precise, those of polynomial nature).