differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

, a branch of mathematics, a smooth functor is a type of

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

defined on finite-dimensional

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

s. Intuitively, a smooth functor is

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

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 bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...

s. Let Vect be the

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

of finite-dimensional

s whose morphisms consist of all

linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

s, 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_(T,U) \rightarrow \mathrm_(F(T),F(U)),

where Hom is notation for

Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory ...

. If this map is

as a map of infinitely

differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

s then ''F'' is said to be a smooth functor. Common smooth functors include, for some vector space ''W'': :''F''(''W'') = ⊗^''n''''W'', the ''n''th iterated

tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...

; :''F''(''W'') = Λ^''n''(''W''), the ''n''th

exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

; and :''F''(''W'') = Sym^''n''(''W''), the ''n''th

symmetric power In mathematics, the ''n''-th symmetric power of an object ''X'' is the quotient of the ''n''-fold product X^n:=X \times \cdots \times X by the permutation action of the symmetric group \mathfrak_n. More precisely, the notion exists at least in the ...

. Smooth functors are significant because any smooth functor can be applied fiberwise to a differentiable

on a manifold. Smoothness of the functor is the condition required to ensure that the patching data for the bundle are smooth as mappings of manifolds. For instance, because the ''n''th exterior power of a vector space defines a smooth functor, the ''n''th 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 space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...

s and vector bundles on infinite-dimensional

Fréchet manifold In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space. More precisely, a Fréchet manifold consists of a Haus ...

s. have developed an infinite-dimensional theory for so-called "

convenient vector space In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. B ...

s" – a class of

locally convex space In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...

s that includes

Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...

Notes

References

*. *. *. {{Functors Functors

See also

Notes

References