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In [[mathematics]], '''Brown's representability theorem''' in [[homotopy theory]]<ref>{{Citation | last1=Switzer | first1=Robert M. | title=Algebraic topology---homotopy and homology | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Classics in Mathematics | isbn=978-3-540-42750-6 | mr=1886843 | year=2002}}, see pages 152–157</ref> gives [[necessary and sufficient condition]]s for a [[contravariant functor]] ''F'' on the [[homotopy category]] ''Hotc'' of pointed connected [[CW complex]]es, to the [[category of sets]] '''Set''', to be a [[representable functor]].


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More specifically, we are given
 
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:''F'': ''Hotc''<sup>op</sup> → '''Set''',


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and there are certain obviously necessary conditions for ''F'' to be of type ''Hom''(&mdash;, ''C''), with ''C'' a pointed connected CW-complex that can be deduced from [[category theory]] alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of [[pointed set]]s; in other words the sets are also given a base point.
 
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==Brown representability theorem for CW complexes==


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The representability theorem for CW complexes, due to E. H. Brown,<ref>{{Citation | last1=Brown | jstor=1970209| first1=Edgar H. | title=Cohomology theories | mr=0138104  | year=1962 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=75 | pages=467–484}}</ref> is the following. Suppose that:
 
# The functor ''F'' maps [[coproducts]] (i.e. [[wedge sum]]s) in ''Hotc'' to products in ''Set'': <math>F(\vee_\alpha X_\alpha) \cong \prod_\alpha F(X_\alpha),</math>
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# The functor ''F'' maps [[Mapping cylinder|homotopy pushouts]] in ''Hotc'' to [[Pullback (category theory)#Weak_pullbacks|weak pullbacks]]. This is often stated as a [[Mayer-Vietoris sequence|Mayer-Vietoris]] axiom: for any CW complex ''W'' covered by two subcomplexes ''U'' and ''V'', and any elements ''u'' ∈ ''F''(''U''), ''v'' ∈ ''F''(''V'') such that ''u'' and ''v'' restrict to the same element of ''F''(''U'' ∩ ''V''), there is an element ''w'' ∈ ''F''(''W'') restricting to ''u'' and ''v'', respectively.
 
Then ''F'' is representable by some CW complex ''C'', that is to say there is an isomorphism
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:''F''(''Z'') &cong; ''Hom''<sub>''Hotc''</sub>(''Z'', ''C'')
 
for any CW complex ''Z'', which is [[Natural transformation|natural]] in ''Z'' in that for any morphism from ''Z'' to another CW complex ''Y'' the induced maps ''F''(''Y'') → ''F''(''Z'') and ''Hom''<sub>''Hot''</sub>(''Y'', ''C'') → ''Hom''<sub>''Hot''</sub>(''Z'', ''C'') are compatible with these isomorphisms.
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The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.


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The representing object ''C'' above can be shown to depend functorially on ''F'': any [[natural transformation]] from ''F'' to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of [[Yoneda's lemma]].
 
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Taking ''F''(''X'') to be the [[singular cohomology]] group ''H''<sup>''i''</sup>(''X'',''A'') with coefficients in a given abelian group ''A'', for fixed ''i'' > 0; then the representing space for ''F'' is the [[Eilenberg-MacLane space]] ''K''(''A'', ''i''). This gives a means of showing the existence of Eilenberg-MacLane spaces.


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Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the [[weak homotopy equivalence]]s, the theorem can equivalently be stated for functors on a category defined in this way.


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However, the theorem is false without the restriction to ''connected'' pointed spaces, and an analogous statement for unpointed spaces is also false.<ref>{{Citation | last1=Freyd | first1=Peter | last2=Heller | first1=Alex | title=Splitting homotopy idempotents. II. | journal=[[Journal of Pure and Applied Algebra]] | volume=89 | year=1993 | pages=93–106 }}</ref>
 
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A similar statement does, however, hold for [[spectrum (homotopy theory)|spectra]] instead of CW complexes.  Brown also proved a general categorical version of the representability theorem,<ref>{{Citation | last1=Brown | first1=Edgar H. | title=Abstract homotopy theory | year=1965 | journal=[[Transactions of the AMS]] | volume=119 | issue=1| pages=79–85 | url=http://www.ams.org/journals/tran/1965-119-01/S0002-9947-1965-0182970-6/}}</ref> which includes both the version for pointed connected CW complexes and the version for spectra.


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A version of the representability theorem in the case of [[triangulated category|triangulated categories]] is due to Amnon Neeman.<ref>{{Citation | last1=Neeman | first1=Amnon | title=The Grothendieck duality theorem via Bousfield's techniques and Brown representability | url=http://www.ams.org/jams/1996-9-01/S0894-0347-96-00174-9/home.html | mr=1308405  | year=1996 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=9 | issue=1 | pages=205–236 | doi=10.1090/S0894-0347-96-00174-9}}</ref> Together with the preceding remark, it gives a criterion for a (covariant) functor ''F'': ''C'' → ''D'' between triangulated categories satisfying certain technical conditions to have a right [[adjoint functor]]. Namely, if ''C'' and ''D'' are triangulated categories with ''C'' compactly generated and ''F'' a triangulated functor commuting with arbitrary direct sums, then ''F'' is a left adjoint. Neeman has applied this to proving the [[Grothendieck duality theorem]] in algebraic geometry.
 
 
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[[Jacob Lurie]] has proved a version of the Brown representability theorem<ref>{{Citation | last1=Lurie |  first1=Jacob | title=Higher Algebra  | year=2011 | url=http://math.harvard.edu/~lurie/papers/higheralgebra.pdf}}</ref> for the homotopy category of a pointed [[quasicategory]] with a compact set of generators which are cogroup objects in the homotopy category. For instance, this applies to the homotopy category of pointed connected CW complexes, as well as to the unbounded [[derived category]] of a Grothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category).
 
 
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[[Category:Homotopy theory]]
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[[Category:Category theory]]
 
[[Category:Representable functors]]
</ul>

Revision as of 06:44, 10 December 2013

In mathematics, Brown's representability theorem in homotopy theory[1] gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor.

More specifically, we are given

F: HotcopSet,

and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.

Brown representability theorem for CW complexes

The representability theorem for CW complexes, due to E. H. Brown,[2] is the following. Suppose that:

  1. The functor F maps coproducts (i.e. wedge sums) in Hotc to products in Set: F(αXα)αF(Xα),
  2. The functor F maps homotopy pushouts in Hotc to weak pullbacks. This is often stated as a Mayer-Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements uF(U), vF(V) such that u and v restrict to the same element of F(UV), there is an element wF(W) restricting to u and v, respectively.

Then F is representable by some CW complex C, that is to say there is an isomorphism

F(Z) ≅ HomHotc(Z, C)

for any CW complex Z, which is natural in Z in that for any morphism from Z to another CW complex Y the induced maps F(Y) → F(Z) and HomHot(Y, C) → HomHot(Z, C) are compatible with these isomorphisms.

The converse statement also holds: any functor represented by a CW complex satisfies the above two properties. This direction is an immediate consequence of basic category theory, so the deeper and more interesting part of the equivalence is the other implication.

The representing object C above can be shown to depend functorially on F: any natural transformation from F to another functor satisfying the conditions of the theorem necessarily induces a map of the representing objects. This is a consequence of Yoneda's lemma.

Taking F(X) to be the singular cohomology group Hi(X,A) with coefficients in a given abelian group A, for fixed i > 0; then the representing space for F is the Eilenberg-MacLane space K(A, i). This gives a means of showing the existence of Eilenberg-MacLane spaces.

Variants

Since the homotopy category of CW-complexes is equivalent to the localization of the category of all topological spaces at the weak homotopy equivalences, the theorem can equivalently be stated for functors on a category defined in this way.

However, the theorem is false without the restriction to connected pointed spaces, and an analogous statement for unpointed spaces is also false.[3]

A similar statement does, however, hold for spectra instead of CW complexes. Brown also proved a general categorical version of the representability theorem,[4] which includes both the version for pointed connected CW complexes and the version for spectra.

A version of the representability theorem in the case of triangulated categories is due to Amnon Neeman.[5] Together with the preceding remark, it gives a criterion for a (covariant) functor F: CD between triangulated categories satisfying certain technical conditions to have a right adjoint functor. Namely, if C and D are triangulated categories with C compactly generated and F a triangulated functor commuting with arbitrary direct sums, then F is a left adjoint. Neeman has applied this to proving the Grothendieck duality theorem in algebraic geometry.

Jacob Lurie has proved a version of the Brown representability theorem[6] for the homotopy category of a pointed quasicategory with a compact set of generators which are cogroup objects in the homotopy category. For instance, this applies to the homotopy category of pointed connected CW complexes, as well as to the unbounded derived category of a Grothendieck abelian category (in view of Lurie's higher-categorical refinement of the derived category).

References

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    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
  5. 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
  6. 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