Nephroid - Revision history

5.68.41.67: /* The evolute... */ added further clarification

2014-08-20T16:14:34Z

The evolute...: added further clarification

← Older revision		Revision as of 18:14, 20 August 2014
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	In [[mathematics]], in the area of [[category theory]], a '''forgetful [[functor]]''' (also known as a '''stripping functor''') 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an [[algebraic structure\|algebraic]] [[structure (mathematical logic)\|structure]] of a given [[signature (logic)\|signature]], this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an [[empty list]], the functor is ~~simply to take the '''underlying set''' of a structure~~. ~~Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set~~ is ~~the most common case.~~		Hello and welcome. My title is Figures Wunder. Hiring is my profession. To gather coins is a thing that I'm totally addicted to. My family members life in Minnesota and my family loves it.<br><br>my web site ... [http://www.animecontent.com/user/ACheongCh www.animecontent.com]

	~~==Introduction==~~

	As examples, there are several forgetful functors from the [[category of commutative rings]]. A ([[unital algebra\|unital]]) [[Ring (mathematics)\|ring]], described in the language of [[universal algebra]], is an ordered tuple (''R'',+,,''a'',0,1) satisfying certain axioms, where "+" and "" are binary functions on the set ''R'', ''a'' is a unary operation corresponding to additive inverse, and 0 and 1 are nullary operations giving the identities of the two binary operations. Deleting the 1 gives a forgetful functor to the category of rings without unit; it simply "forgets" the unit. Deleting "*" and 1 yields a functor to the category of [[abelian group]]s, which assigns to each ring ''R'' the underlying additive abelian group of ''R''. To ~~each [[morphism]] of rings~~ is ~~assigned the same [[function (mathematics)\|function]] considered merely as~~ a ~~morphism of addition between the underlying groups. Deleting all the operations gives the functor to the underlying set ''R''.~~

	~~It is beneficial to distinguish between forgetful functors~~ that "forget structure" versus those that "forget properties". For example, in the above example of commutative rings, in addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the category '''CRing''' to '''Ring''' ~~that forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, which part~~ to ~~consider the underlying set is a matter of taste, though this is rarely ambiguous in practice). For these objects, there are forgetful functors that forget the extra sets that are more general~~.

	~~Most common objects studied~~ in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc.) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows.
	Let <math>\mathcal{C}</math> be any category based on [[Set (mathematics)\|sets]], e.g. [[group (mathematics)\|group]]s - sets of elements - or [[topological space]]s - sets of 'points'. As usual, write <math>\operatorname{Ob}(\mathcal{C})</math> for the [[Category theory#Categories.2C objects.2C and ~~morphisms\|objects]] of <math>\mathcal{C}</math> and write <math>\operatorname{Fl}(\mathcal{C})</math> for the morphisms of the same~~. ~~Consider the rule:~~
	~~:<math>A</math> in <math>\operatorname{Ob}(\mathcal{C}),\quad A\mapsto \|A\|=</math> the underlying set of <math>A,</math>~~
	~~:<math>u~~<~~/math~~> in <~~math~~>~~\operatorname{Fl}(\mathcal{C}),\quad u\mapsto \|u\|=</math> the morphism, <math>u</math>, as a map of sets~~.
	~~The functor <math>\|\;\;\|</math> is then the forgetful functor from <math>\mathcal{C}</math> to <math>\mathbf{Set}</math>, the [[category of sets]]~~.

	~~Forgetful functors are almost always [[faithful functor\|faithful]]~~. [~~[Concrete category\|Concrete categories]] have forgetful functors to the category of sets—indeed they may be ''defined'' as those categories that admit a faithful functor to that category.~~

	Forgetful functors that only forget axioms are always fully faithful; every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms. Forgetful functors that forget structures need not be full; some morphisms don't respect the structure. These functors are still faithful though; distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.

	~~In the language of formal logic, a functor of the first kind removes axioms. The second kind removes predicates. The third kind remove types.~~

	An example of the first kind is the forgetful functor '''Ab''' → '''Grp'''. One of the second kind is the forgetful functor '''Ab''' → '''Set'''. A functor of the third kind is the functor '''Mod''' → '''Ab''', where '''Mod''' is the [[fibred category]] of all modules over arbitrary rings. To see this, just choose a ring homomorphism between the underlying rings that does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object in '''Mod''' is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste.

	~~==Left Adjoint: Free==~~
	~~Forgetful functors tend to have [[left adjoint]]s, which are '[[free object\|free]]' constructions. For example:~~
	* [[free module]]: the forgetful functor from <math>\mathbf{Mod}(R)</math> (the category of <math>R</math>-[[module (mathematics)\|module]]) to <math>\mathbf{Set}</math> has left adjoint <math>\operatorname{Free}_R</math>, with <math>X\mapsto \operatorname{Free}_R(X)</math>, the free <math>R</math>-module with [[Basis (linear algebra)\|basis]] <math>X</math>.
	* [[free group]]
	* [[free lattice]]
	* [[tensor algebra]]
	* [[free category]], adjoint to the forgetful functor from categories to [[quiver (mathematics)\|quivers]]

	~~For a more extensive list, see (Mac Lane 1997).~~

	~~As this is a fundamental example of adjoints, we spell it out~~:
	~~adjointness means that given a set ''X'' and an object (say, an ''R''-module) ''M'', maps ''of sets'' <math>X \to M<~~/~~math> correspond to maps of modules <math>\operatorname{Free}_R(X) \to M<~~/~~math>: every map of sets yields a map of modules, and every map of modules comes from a map of sets.~~

	~~In the case of vector spaces, this is summarized as:~~
	~~"A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything~~."

	~~Symbolically:~~
	~~:<math>\operatorname{Hom}_{\mathbf{Mod}_R}(\operatorname{Free}_R(X),M) = \operatorname{Hom}_{\mathbf{Set}}(X,\operatorname{Forget}(M))~~.</~~math>~~

	~~The [[Free_object#Free_functor\|unit of the free-forget adjunction]] is the "inclusion of a basis": <math>X \to \operatorname{Free}_R(X)<~~/~~math>~~.

	~~'''Fld''', the category of fields, furnishes an example of a forgetful functor with no adjoint~~. ~~There is no field satisfying a free universal property for a given set.~~

	~~==References==~~
	* [[Saunders Mac Lane\|Mac Lane, Saunders]]. ''Categories for the Working Mathematician'', Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1997. ISBN 0-387-98403-8
	* {{nlab\|id=forgetful+functor\|title=Forgetful functor}}

	~~{{Functors}}~~

	~~{{DEFAULTSORT:Forgetful Functor}}~~
	~~[[Category:Functors]~~]

en>Clive tooth: All the rays are in blue and the incoming rays do not intersect the circle at points 4 degrees apart.

2014-01-27T22:09:53Z

All the rays are in blue and the incoming rays do not intersect the circle at points 4 degrees apart.

← Older revision		Revision as of 00:09, 28 January 2014
Line 1:		Line 1:
	~~It is quite frustrating~~ to ~~struggle for getting pregnant without any results~~. * One of the ~~best and proven tips for getting pregnant~~ in a ~~natural way~~ is ~~to have sex at least three times a week~~. ~~To work out when you~~ are ~~ovulating~~, ~~find days 12 – 16~~ in ~~your cycle. These vitamins~~ are ~~especially equipped~~ to ~~provide your body~~, and ~~baby~~, ~~with~~ the ~~nutrients they need~~ to ~~get off~~ to ~~a head start~~. ~~To make it clear once more~~, ~~you did not know you had Lyme while you were pregnant. <br><br>If youre trying~~ to ~~get pregnant and youre not successful~~, there ~~could be~~ ~~some causes~~ that ~~reduce~~ the ~~chances~~ of ~~pregnancy:~~. ~~need to have prior to travelling~~ to a ~~certain country~~. ~~That is why you need to make use~~ of ~~such thermometer~~. ~~You can direct Sims to fall in love and start their very own family~~. ~~com where you can ask questions~~, ~~get information~~, or ~~just let off a bit~~ of ~~steam~~. <br><br>~~There are ovulation prediction kits~~ and ~~saliva tests available~~ and ~~you can get them over~~ the ~~counter at your local pharmacy~~. ~~Sperms count number greater considerably just after~~ a ~~couple~~ of ~~therapies~~. ~~Studies show~~ that ~~people who~~ do ~~these things have easier pregnancies and healthier babies~~. ~~Irrespective~~ of the ~~fact that I was in my late 30's about to enter into my early 40's~~, a ~~breakthrough appeared in my life in~~ the ~~form~~ of ~~a wonder book on pregnancy~~. ~~In this case~~ the ~~egg~~ is ~~normal~~, ~~but for what ever reason~~, the ~~sperm can~~ not ~~penetrate it~~. ~~<br><br>Most of~~ the ~~battle~~ is ~~usually over if one has~~ a ~~male partner~~ and a ~~willing sperm donor~~. ~~I hope you found the above information on~~ [~~http~~://~~www.quebec~~-~~1759.info~~/~~sitemap~~/ ~~how to get pregnant~~ with ~~twins~~] ~~naturally to be helpful~~. ~~Learn about how your body works~~, ~~about what works in your favor and how~~ to ~~read your fertile signs~~. ~~Stress~~ is a ~~factor~~ that ~~can negatively affect fertility~~. ~~To begin~~ the ~~process~~, ~~you must have a verified pregnancy test done~~ by a ~~clinic~~, ~~doctor, or other place that says they give verification~~. <br><br>- ~~Vitamin C~~ is ~~an essential nutrient for humans in which it functions as an anti-oxidant, since it protects~~ the ~~body against oxidative stress It is also~~ a ~~cofactor in at least eight enzymatic reactions~~, ~~including several collagen synthesis reactions that cause~~ the ~~most severe symptoms~~ of ~~scurvy when they are dysfunctional. Most~~ of ~~the infertility treatments undertaken by doctors have~~ a ~~list of possible side effects~~. ~~Method 1 – Determining how to ‘read’ Cervical Fluid as~~ a ~~sign of~~ the ~~Best Time To Get Pregnant. Merely performing intercourse isn~~'~~t necessarily all there is to getting pregnant. My oldest daughter had some issues~~ in ~~high school with her joints and she was unable to do some of her cheer leading moves because of it~~, ~~but for the most part she was and is functioning normally~~.		In [[mathematics]], in the area of [[category theory]], a '''forgetful [[functor]]''' (also known as a '''stripping functor''') 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an [[algebraic structure\|algebraic]] [[structure (mathematical logic)\|structure]] of a given [[signature (logic)\|signature]], this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an [[empty list]], the functor is simply to take the '''underlying set''' of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.

			==Introduction==

			As examples, there are several forgetful functors from the [[category of commutative rings]]. A ([[unital algebra\|unital]]) [[Ring (mathematics)\|ring]], described in the language of [[universal algebra]], is an ordered tuple (''R'',+,,''a'',0,1) satisfying certain axioms, where "+" and "" are binary functions on the set ''R'', ''a'' is a unary operation corresponding to additive inverse, and 0 and 1 are nullary operations giving the identities of the two binary operations. Deleting the 1 gives a forgetful functor to the category of rings without unit; it simply "forgets" the unit. Deleting "*" and 1 yields a functor to the category of [[abelian group]]s, which assigns to each ring ''R'' the underlying additive abelian group of ''R''. To each [[morphism]] of rings is assigned the same [[function (mathematics)\|function]] considered merely as a morphism of addition between the underlying groups. Deleting all the operations gives the functor to the underlying set ''R''.

			It is beneficial to distinguish between forgetful functors that "forget structure" versus those that "forget properties". For example, in the above example of commutative rings, in addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the category '''CRing''' to '''Ring''' that forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, which part to consider the underlying set is a matter of taste, though this is rarely ambiguous in practice). For these objects, there are forgetful functors that forget the extra sets that are more general.

			Most common objects studied in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc.) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows.
			Let <math>\mathcal{C}</math> be any category based on [[Set (mathematics)\|sets]], e.g. [[group (mathematics)\|group]]s - sets of elements - or [[topological space]]s - sets of 'points'. As usual, write <math>\operatorname{Ob}(\mathcal{C})</math> for the [[Category theory#Categories.2C objects.2C and morphisms\|objects]] of <math>\mathcal{C}</math> and write <math>\operatorname{Fl}(\mathcal{C})</math> for the morphisms of the same. Consider the rule:
			:<math>A</math> in <math>\operatorname{Ob}(\mathcal{C}),\quad A\mapsto \|A\|=</math> the underlying set of <math>A,</math>
			:<math>u</math> in <math>\operatorname{Fl}(\mathcal{C}),\quad u\mapsto \|u\|=</math> the morphism, <math>u</math>, as a map of sets.
			The functor <math>\|\;\;\|</math> is then the forgetful functor from <math>\mathcal{C}</math> to <math>\mathbf{Set}</math>, the [[category of sets]].

			Forgetful functors are almost always [[faithful functor\|faithful]]. [[Concrete category\|Concrete categories]] have forgetful functors to the category of sets—indeed they may be ''defined'' as those categories that admit a faithful functor to that category.

			Forgetful functors that only forget axioms are always fully faithful; every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms. Forgetful functors that forget structures need not be full; some morphisms don't respect the structure. These functors are still faithful though; distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.

			In the language of formal logic, a functor of the first kind removes axioms. The second kind removes predicates. The third kind remove types.

			An example of the first kind is the forgetful functor '''Ab''' → '''Grp'''. One of the second kind is the forgetful functor '''Ab''' → '''Set'''. A functor of the third kind is the functor '''Mod''' → '''Ab''', where '''Mod''' is the [[fibred category]] of all modules over arbitrary rings. To see this, just choose a ring homomorphism between the underlying rings that does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object in '''Mod''' is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste.

			==Left Adjoint: Free==
			Forgetful functors tend to have [[left adjoint]]s, which are '[[free object\|free]]' constructions. For example:
			* [[free module]]: the forgetful functor from <math>\mathbf{Mod}(R)</math> (the category of <math>R</math>-[[module (mathematics)\|module]]) to <math>\mathbf{Set}</math> has left adjoint <math>\operatorname{Free}_R</math>, with <math>X\mapsto \operatorname{Free}_R(X)</math>, the free <math>R</math>-module with [[Basis (linear algebra)\|basis]] <math>X</math>.
			* [[free group]]
			* [[free lattice]]
			* [[tensor algebra]]
			* [[free category]], adjoint to the forgetful functor from categories to [[quiver (mathematics)\|quivers]]

			For a more extensive list, see (Mac Lane 1997).

			As this is a fundamental example of adjoints, we spell it out:
			adjointness means that given a set ''X'' and an object (say, an ''R''-module) ''M'', maps ''of sets'' <math>X \to M</math> correspond to maps of modules <math>\operatorname{Free}_R(X) \to M</math>: every map of sets yields a map of modules, and every map of modules comes from a map of sets.

			In the case of vector spaces, this is summarized as:
			"A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything."

			Symbolically:
			:<math>\operatorname{Hom}_{\mathbf{Mod}_R}(\operatorname{Free}_R(X),M) = \operatorname{Hom}_{\mathbf{Set}}(X,\operatorname{Forget}(M)).</math>

			The [[Free_object#Free_functor\|unit of the free-forget adjunction]] is the "inclusion of a basis": <math>X \to \operatorname{Free}_R(X)</math>.

			'''Fld''', the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set.

			==References==
			* [[Saunders Mac Lane\|Mac Lane, Saunders]]. ''Categories for the Working Mathematician'', Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1997. ISBN 0-387-98403-8
			* {{nlab\|id=forgetful+functor\|title=Forgetful functor}}

			{{Functors}}

			{{DEFAULTSORT:Forgetful Functor}}
			[[Category:Functors]]

222.164.140.80: /* ...an envelope of lines */ changed "4" to "4 degrees" for clarity

2012-09-02T01:32:09Z

...an envelope of lines: changed "4" to "4 degrees" for clarity

New page

It is quite frustrating to struggle for getting pregnant without any results. * One of the best and proven tips for getting pregnant in a natural way is to have sex at least three times a week. To work out when you are ovulating, find days 12 – 16 in your cycle. These vitamins are especially equipped to provide your body, and baby, with the nutrients they need to get off to a head start. To make it clear once more, you did not know you had Lyme while you were pregnant. If youre trying to get pregnant and youre not successful, there could be some causes that reduce the chances of pregnancy:. need to have prior to travelling to a certain country. That is why you need to make use of such thermometer. You can direct Sims to fall in love and start their very own family. com where you can ask questions, get information, or just let off a bit of steam. There are ovulation prediction kits and saliva tests available and you can get them over the counter at your local pharmacy. Sperms count number greater considerably just after a couple of therapies. Studies show that people who do these things have easier pregnancies and healthier babies. Irrespective of the fact that I was in my late 30's about to enter into my early 40's, a breakthrough appeared in my life in the form of a wonder book on pregnancy. In this case the egg is normal, but for what ever reason, the sperm can not penetrate it. Most of the battle is usually over if one has a male partner and a willing sperm donor. I hope you found the above information on [http://www.quebec-1759.info/sitemap/ how to get pregnant with twins] naturally to be helpful. Learn about how your body works, about what works in your favor and how to read your fertile signs. Stress is a factor that can negatively affect fertility. To begin the process, you must have a verified pregnancy test done by a clinic, doctor, or other place that says they give verification. - Vitamin C is an essential nutrient for humans in which it functions as an anti-oxidant, since it protects the body against oxidative stress It is also a cofactor in at least eight enzymatic reactions, including several collagen synthesis reactions that cause the most severe symptoms of scurvy when they are dysfunctional. Most of the infertility treatments undertaken by doctors have a list of possible side effects. Method 1 – Determining how to ‘read’ Cervical Fluid as a sign of the Best Time To Get Pregnant. Merely performing intercourse isn't necessarily all there is to getting pregnant. My oldest daughter had some issues in high school with her joints and she was unable to do some of her cheer leading moves because of it, but for the most part she was and is functioning normally.