In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces.
Let Vect be the category of finite-dimensional real vector spaces whose morphisms consist of all linear mappings, and let F be a covariant functor that maps Vect to itself.
[1] Common smooth functors include, for some vector space W:[2] Smooth functors are significant because any smooth functor can be applied fiberwise to a differentiable vector bundle on a manifold.
[2] For instance, because the nth exterior power of a vector space defines a smooth functor, the nth exterior power of a smooth vector bundle is also a smooth vector bundle.
Although there are established methods for proving smoothness of standard constructions on finite-dimensional vector bundles, smooth functors can be generalized to categories of topological vector spaces and vector bundles on infinite-dimensional Fréchet manifolds.