Concept# Smooth functor

Summary

In differential topology, a branch of mathematics, a smooth functor is a type of functor defined on finite-dimensional real vector spaces. Intuitively, a smooth functor is smooth in the sense that it sends smoothly parameterized families of vector spaces to smoothly parameterized families of vector spaces. Smooth functors may therefore be uniquely extended to functors defined on vector bundles.
Let Vect be the 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. For vector spaces T, U ∈ Vect, the functor F induces a mapping
:F : \mathrm{Hom}*{\mathbf{Vect}}(T,U) \rightarrow \mathrm{Hom}*{\mathbf{Vect}}(F(T),F(U)),
where Hom is notation for Hom functor. If this map is smooth as a map of infinitely differentiable manifolds then F is said to be a smooth functor.
Common smooth functors include, for some vector space W:
:F(W) = ⊗nW, the nth iterated tensor product;
:F

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