1985–86 Mersin İdmanyurdu season: Difference between revisions

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In [[differential topology]], a branch of mathematics, a '''smooth functor''' is a type of [[functor]] defined on finite-dimensional [[real number|real]] [[vector space]]s.  Intuitively, a smooth functor is [[infinitely differentiable|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 bundle]]s.


Let '''Vect''' be the [[Category (mathematics)|category]] of finite-dimensional [[Real number|real]] [[vector space]]s whose morphisms consist of all [[linear mapping]]s, and let ''F'' be a [[covariant functor|covariant]] functor that maps '''Vect''' to itself. For vector spaces ''T'', ''U'' ∈ '''Vect''', the functor ''F'' induces a mapping
:<math>F : \mathrm{Hom}_{\mathbf{Vect}}(T,U) \rightarrow \mathrm{Hom}_{\mathbf{Vect}}(F(T),F(U)),</math>
where Hom is notation for [[Hom functor]]. If this map is [[smooth function|smooth]] as a map of infinitely [[differentiable manifold]]s then ''F'' is said to be a '''smooth functor'''.<ref>{{harvnb|Antonelli|2003|p=1420}}; {{harvnb|Kriegl|Michor|1997|p=290}}.  {{harvnb|Lee|2002}}, pp.122–23 defines smooth functors over a different category, whose morphisms are [[linear isomorphism]]s rather than all linear mappings.</ref>


Common smooth functors include, for some vector space ''W'':<ref name=KM290>{{harvnb|Kriegl|Michor|1997|p=290}}</ref>
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:''F''(''W'')&nbsp;=&nbsp;⊗<sup>''n''</sup>''W'', the ''n''th iterated [[tensor product]];
:''F''(''W'')&nbsp;=&nbsp;&Lambda;<sup>''n''</sup>(''W''), the ''n''th [[exterior power]]; and
:''F''(''W'')&nbsp;=&nbsp;Sym<sup>''n''</sup>(''W''), the ''n''th [[symmetric power]].
 
Smooth functors are significant because any smooth functor can be applied fiberwise to a differentiable [[vector bundle]] 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.<ref name=KM290 /> 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]]s and vector bundles on infinite-dimensional [[Fréchet manifold]]s.<ref>{{harvnb|Kriegl|Michor|1997}} have developed an infinite-dimensional theory for so-called "convenient" topological vector spaces – a class of [[locally convex space]]s that includes [[Fréchet space]]s.</ref>
 
==See also==
*[[Smooth infinitesimal analysis]]
*[[Synthetic differential geometry]]
 
== Notes ==
<references/>
 
==References==
 
*{{citation|title=Handbook of Finsler geometry|first=P. L. |last=Antonelli|publisher=Springer|year=2003|isbn=1-4020-1556-9|page=1420}}.
*{{citation|title=The convenient setting of global analysis|first1=Andreas|last1=Kriegl|first2=Peter W.|last2=Michor|publisher=AMS Bookstore|year=1997|isbn=0-8218-0780-3|page=290}}.
*{{citation|title=Introduction to smooth manifolds|first=John M.|last=Lee|publisher=Springer|year=2002|isbn=0-387-95448-1|pages=122–23}}.
 
{{Functors}}
 
[[Category:Functors]]

Latest revision as of 09:40, 16 November 2014


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