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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> | |||
:''F''(''W'') = ⊗<sup>''n''</sup>''W'', the ''n''th iterated [[tensor product]]; | |||
:''F''(''W'') = Λ<sup>''n''</sup>(''W''), the ''n''th [[exterior power]]; and | |||
:''F''(''W'') = 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]] | |||
Revision as of 16:06, 25 January 2014
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 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. For vector spaces T, U ∈ Vect, the functor F induces a mapping
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.[1]
Common smooth functors include, for some vector space W:[2]
- F(W) = ⊗nW, the nth iterated tensor product;
- F(W) = Λn(W), the nth exterior power; and
- F(W) = Symn(W), the nth 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.[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.[3]
See also
Notes
- ↑ Template:Harvnb; Template:Harvnb. Template:Harvnb, pp.122–23 defines smooth functors over a different category, whose morphisms are linear isomorphisms rather than all linear mappings.
- ↑ 2.0 2.1 Template:Harvnb
- ↑ Template:Harvnb have developed an infinite-dimensional theory for so-called "convenient" topological vector spaces – a class of locally convex spaces that includes Fréchet spaces.
References
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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.