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{{For|the notion of duality in algebraic topology|Eckmann–Hilton duality}} | |||
In [[mathematics]], the '''Eckmann–Hilton argument''' (or '''Eckmann–Hilton principle''' or '''Eckmann–Hilton theorem''') is an [[argument]] about two [[monoid]] [[structure (category theory)|structure]]s on a [[Set (mathematics)|set]] where one is a [[homomorphism]] for the other. Given this, the structures can be shown to coincide, and the resulting [[monoid]] demonstrated to be [[commutative]]. This can then be used to prove the commutativity of the higher [[homotopy group]]s. The principle is named after [[Beno Eckmann]] and [[Peter Hilton]], who used it in a 1962 paper. | |||
==The Eckmann–Hilton result== | |||
Let <math>X</math> be a set equipped with two binary operations, which we will write . and *, and suppose: | |||
1. * and . are both [[unital algebra|unital]], with the same unit 1, say, and<br /> | |||
2. <math>\forall a,b,c,d \in X,\ (a*b).(c*d) = (a.c)*(b.d)</math>. | |||
Then * and . are the same and in fact commutative and associative. | |||
==Remarks== | |||
The operations * and . are often referred to as multiplications, but this might imply they are associative, a property which is not required for the proof. In fact, associativity follows; moreover, condition 1 above can be weakened to the assertion that both operations are unital, since it can be proved from condition 2 that the units must then coincide. If the operations are associative, each one defines the structure of a monoid on <math>X</math>, and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism <math>(X,.)\times(X,.)\to(X,.)</math> (or vice versa). An even more abstract way of stating the theorem is: If <math>X</math> is a monoid object in the category of monoids, then <math>X</math> is in fact a commutative monoid. | |||
It is important that a similar argument does NOT give such a triviality result in the case of monoid objects in the categories of small categories or of groupoids. Instead the notion of group object in the category of [[groupoid]]s turns out to be equivalent to the notion of [[crossed module]]. This leads to the idea of using multiple groupoid objects in homotopy theory. | |||
More generally, the Eckmann–Hilton argument is a special case of the use of the [[interchange law]] in the theory of (strict) double and multiple categories. A (strict) [[double category]] is a set, or class, equipped with two category structures, each of which is a morphism for the other structure. If the compositions in the two category structures are written <math>\circ, \bullet</math> then the interchange law reads | |||
:<math> (a \circ b)\bullet (c \circ d) = (a \bullet c) \circ (b \bullet d)</math> | |||
whenever both sides are defined. For an example of its use, and some discussion, see the paper of Higgins referenced below. The interchange law implies that a double category contains a family of abelian monoids. | |||
The history in relation to [[homotopy group]]s is interesting.{{Says who|date=May 2013}} The workers in topology of the early 20th century were aware that the nonabelian fundamental group was of use in geometry and analysis; that abelian homology groups could be defined in all dimensions; and that for a connected space, the first homology group was the fundamental group made abelian. So there was a desire to generalise the nonabelian fundamental group to all dimensions. | |||
In 1932, E. [[Cech]] submitted a paper on higher homotopy groups to the International Congress of Mathematics at Zurich. However, Alexandroff and Hopf quickly proved these groups were abelian for <math> n > 1</math>, and on these grounds persuaded Cech to withdraw his paper, so that only a small paragraph appeared in the ''Proceedings''. It is said that Hurewicz attended this conference, and his first work on higher homotopy groups appeared in 1935.{{cn|date=May 2013}} Thus the dreams of the early topologists have long been regarded as a mirage.{{cn|date=May 2013}} | |||
Cubical higher homotopy groupoids are constructed for filtered spaces in the book ''[http://pages.bangor.ac.uk/~mas010/nonab-a-t.html Nonabelian algebraic topology]'' cited below, which develops basic algebraic topology, including higher Seifert van Kampen Theorems, without using [[singular homology]] or simplicial approximation. | |||
==Proof== | |||
The proof is not hard, although it is much more conceptually clear if geometric diagrams are used. In ordinary algebra notation, the proof is as follows: | |||
Let <math>a,b \in X</math>. Then <math> a.b = (1*a).(b*1) = (1.b)*(a.1) = b*a = (b.1)*(1.a) = (b*1).(1*a) = b.a \,</math> | |||
It is sometimes represented as a 'clock'. In this image, "0" is the unit for p⊕q and "1" is the unit for ⊗. Starting from any position on the clock we can move to the next by some use of the unital character of "0" and "1" or the distributive rule: | |||
[[Image:eckman hilton clock.svg|400px|center]] | |||
==References== | |||
*[http://math.ucr.edu/home/baez/week89.html John Baez: Eckmann–Hilton principle (week 89)] | |||
*[http://math.ucr.edu/home/baez/week100.html John Baez: Eckmann–Hilton principle (week 100)] | |||
*{{citation | |||
| last1 = Eckmann | first1 = B. | |||
| last2 = Hilton | first2 = P. J. | |||
| mr = 0136642 | |||
| journal = [[Mathematische Annalen]] | |||
| pages = 227–255 | |||
| title = Group-like structures in general categories. I. Multiplications and comultiplications | |||
| volume = 145 | |||
| year = 1962}}. | |||
*{{citation | |||
| last1 = Brown | first1 = R. | |||
| last2 = Higgins | first2 = P. J. | |||
| last3 = Sivera | first3 = R. | |||
| mr = 2841564 | |||
| series = [[European Mathematical Society]] Tracts in Mathematics | |||
| pages = 703 | |||
| title = Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids | |||
| volume = 15 | |||
| year = 2011}}. | |||
*{{citation | |||
| last1 = Higgins | first2 = P. J. | |||
| mr = 2122826 | |||
| journal = [http://www.tac.mta.ca/tac/volumes/14/4/14-04abs.html Theory and Application of Categories] | |||
| pages = 60–74 | |||
| title = Thin elements and commutative shells in cubical $\omega$-categories | |||
| volume = 14 | |||
| year = 2005}}. | |||
==External links== | |||
*[http://www.youtube.com/watch?v=Rjdo-RWQVIY Eugenia Cheng of 'the Catsters' video team explains the Eckmann Hilton argument.] | |||
* [http://pages.bangor.ac.uk/~mas010/hdaweb2.htm Higher dimensional group theory] | |||
{{DEFAULTSORT:Eckmann-Hilton argument}} | |||
[[Category:Category theory]] | |||
Revision as of 06:25, 15 January 2014
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance.
In mathematics, the Eckmann–Hilton argument (or Eckmann–Hilton principle or Eckmann–Hilton theorem) is an argument about two monoid structures on a set where one is a homomorphism for the other. Given this, the structures can be shown to coincide, and the resulting monoid demonstrated to be commutative. This can then be used to prove the commutativity of the higher homotopy groups. The principle is named after Beno Eckmann and Peter Hilton, who used it in a 1962 paper.
The Eckmann–Hilton result
Let be a set equipped with two binary operations, which we will write . and *, and suppose:
1. * and . are both unital, with the same unit 1, say, and
2. .
Then * and . are the same and in fact commutative and associative.
Remarks
The operations * and . are often referred to as multiplications, but this might imply they are associative, a property which is not required for the proof. In fact, associativity follows; moreover, condition 1 above can be weakened to the assertion that both operations are unital, since it can be proved from condition 2 that the units must then coincide. If the operations are associative, each one defines the structure of a monoid on , and the conditions above are equivalent to the more abstract condition that * is a monoid homomorphism (or vice versa). An even more abstract way of stating the theorem is: If is a monoid object in the category of monoids, then is in fact a commutative monoid.
It is important that a similar argument does NOT give such a triviality result in the case of monoid objects in the categories of small categories or of groupoids. Instead the notion of group object in the category of groupoids turns out to be equivalent to the notion of crossed module. This leads to the idea of using multiple groupoid objects in homotopy theory.
More generally, the Eckmann–Hilton argument is a special case of the use of the interchange law in the theory of (strict) double and multiple categories. A (strict) double category is a set, or class, equipped with two category structures, each of which is a morphism for the other structure. If the compositions in the two category structures are written then the interchange law reads
whenever both sides are defined. For an example of its use, and some discussion, see the paper of Higgins referenced below. The interchange law implies that a double category contains a family of abelian monoids.
The history in relation to homotopy groups is interesting.Template:Says who The workers in topology of the early 20th century were aware that the nonabelian fundamental group was of use in geometry and analysis; that abelian homology groups could be defined in all dimensions; and that for a connected space, the first homology group was the fundamental group made abelian. So there was a desire to generalise the nonabelian fundamental group to all dimensions.
In 1932, E. Cech submitted a paper on higher homotopy groups to the International Congress of Mathematics at Zurich. However, Alexandroff and Hopf quickly proved these groups were abelian for , and on these grounds persuaded Cech to withdraw his paper, so that only a small paragraph appeared in the Proceedings. It is said that Hurewicz attended this conference, and his first work on higher homotopy groups appeared in 1935.Template:Cn Thus the dreams of the early topologists have long been regarded as a mirage.Template:Cn
Cubical higher homotopy groupoids are constructed for filtered spaces in the book Nonabelian algebraic topology cited below, which develops basic algebraic topology, including higher Seifert van Kampen Theorems, without using singular homology or simplicial approximation.
Proof
The proof is not hard, although it is much more conceptually clear if geometric diagrams are used. In ordinary algebra notation, the proof is as follows:
It is sometimes represented as a 'clock'. In this image, "0" is the unit for p⊕q and "1" is the unit for ⊗. Starting from any position on the clock we can move to the next by some use of the unital character of "0" and "1" or the distributive rule:
References
- John Baez: Eckmann–Hilton principle (week 89)
- John Baez: Eckmann–Hilton principle (week 100)
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- 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.
- 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.