Polynomial functor

In algebra, a polynomial functor is an endofunctor on the category

of finite-dimensional vector spaces that depends polynomially on vector spaces.

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

over a field of characteristic zero.

[1] Let k be a field of characteristic zero and

the category of finite-dimensional k-vector spaces and k-linear maps.

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

with common domain and codomain, the vector-valued polynomial

is homogeneous of degree n. If “finite vector spaces” is replaced by “finite sets”, one gets the notion of combinatorial species (to be precise, those of polynomial nature).

This category theory-related article is a stub.