Smooth functor: Difference between revisions
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In [[category theory]], a '''smooth functor''' is a type of [[functor]]. Let it be supposed that, given a linear-[[isomorphism|isomorphic]] [[category]] ''V'' of finite-dimensional [[real]] [[vector space]]s, there is a [[covariant functor|covariant]] functor ''F'' that maps ''V'' to itself. For vector spaces |
In [[category theory]], a branch of mathematics, a '''smooth functor''' is a type of [[functor]]. Let it be supposed that, given a linear-[[isomorphism|isomorphic]] [[category]] ''V'' of finite-dimensional [[real]] [[vector space]]s, there is a [[covariant functor|covariant]] functor ''F'' that maps ''V'' to itself. For vector spaces |
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:<math>T,U \isin \mathrm{Ob}(V),</math> |
:<math>T,U \isin \mathrm{Ob}(V),</math> |
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there is a mapping |
there is a mapping |
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Revision as of 22:58, 16 July 2009
In category theory, a branch of mathematics, a smooth functor is a type of functor. Let it be supposed that, given a linear-isomorphic category V of finite-dimensional real vector spaces, there is a covariant functor F that maps V to itself. For vector spaces
there is a mapping
where Hom is notation for Hom functor. If this map is smooth – as smooth as maps between smooth (or infinitely differentiable) manifolds[1] – F is said to be a smooth functor.[1][2][3] Common smooth functors include, for some vector space W:[2]
- – the nth iterated convenient tensor product,
- – the nth exterior product, and
Citations
- ^ a b P. L. Antonelli (2003). Handbook of Finsler geometry. Springer. p. 1420. ISBN 1402015569.
- ^ a b A. Kriegl (1997). The convenient setting of global analysis. AMS Bookstore. p. 290. ISBN 0821807803.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - ^ J. M. Lee (2002). Introduction to smooth manifolds. Springer. pp. 122–23. ISBN 0387954481.