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en>HighKing: /* Open-source projects */ cleanup

2014-01-09T21:36:12Z

Open-source projects: cleanup

← Older revision		Revision as of 23:36, 9 January 2014
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	Hi there! :) ~~My name~~ is ~~Isidro~~, I'm a ~~student studying Educational Studies~~ from ~~Fredericia~~, ~~Denmark~~.<br><br>~~My homepage~~ ... ~~Crazy Taxi City Rush Hack~~ ([~~https~~://~~Www~~.~~Facebook~~.~~com~~/~~CrazyTaxiCityRushHackToolCheatsAndroidiOS click through~~ the ~~up coming article~~])		In [[mathematics]], particularly [[category theory]], a '''representable functor''' is a [[functor]] of a special form from an arbitrary [[category (mathematics)\|category]] into the [[category of sets]]. Such functors give representations of an abstract category in terms of known structures (i.e. [[Set (mathematics)\|sets]] and [[function (mathematics)\|function]]s) allowing one to utilize, as much as possible, knowledge about the category of sets in other settings.

			From another point of view, representable functors for a category ''C'' are the functors ''given'' with ''C''. Their theory is a vast generalisation of [[upper set]]s in [[poset]]s, and of [[Cayley's theorem]] in [[group theory]].

			==Definition==

			Let '''C''' be a [[locally small category]] and let '''Set''' be the [[category of sets]]. For each object ''A'' of '''C''' let Hom(''A'',–) be the [[hom functor]] which maps objects ''X'' to the set Hom(''A'',''X'').

			A [[functor]] ''F'' : '''C''' → '''Set''' is said to be '''representable''' if it is [[naturally isomorphic]] to Hom(''A'',–) for some object ''A'' of '''C'''. A '''representation''' of ''F'' is a pair (''A'', Φ) where
			:Φ : Hom(''A'',–) → ''F''
			is a natural isomorphism.

			A [[contravariant functor]] ''G'' from '''C''' to '''Set''' is the same thing as a functor ''G'' : '''C'''<sup>op</sup> → '''Set''' and is commonly called a [[presheaf (category theory)\|presheaf]]. A presheaf is representable when it is naturally isomorphic to the contravariant hom-functor Hom(–,''A'') for some object ''A'' of '''C'''.

			==Universal elements==

			According to [[Yoneda's lemma]], natural transformations from Hom(''A'',–) to ''F'' are in one-to-one correspondence with the elements of ''F''(''A''). Given a natural transformation Φ : Hom(''A'',–) → ''F'' the corresponding element ''u'' ∈ ''F''(''A'') is given by
			:<math>u = \Phi_A(\mathrm{id}_A).\,</math>
			Conversely, given any element ''u'' ∈ ''F''(''A'') we may define a natural transformation Φ : Hom(''A'',–) → ''F'' via
			:<math>\Phi_X(f) = (Ff)(u)\,</math>
			where ''f'' is an element of Hom(''A'',''X''). In order to get a representation of ''F'' we want to know when the natural transformation induced by ''u'' is an isomorphism. This leads to the following definition:
			:A '''universal element''' of a functor ''F'' : '''C''' → '''Set''' is a pair (''A'',''u'') consisting of an object ''A'' of '''C''' and an element ''u'' ∈ ''F''(''A'') such that for every pair (''X'',''v'') with ''v'' ∈ ''F''(''X'') there exists a unique morphism ''f'' : ''A'' → ''X'' such that (''Ff'')''u'' = ''v''.
			A universal element may be viewed as a [[universal morphism]] from the one-point set {•} to the functor ''F'' or as an [[initial object]] in the [[category of elements]] of ''F''.

			The natural transformation induced by an element ''u'' ∈ ''F''(''A'') is an isomorphism if and only if (''A'',''u'') is a universal element of ''F''. We therefore conclude that representations of ''F'' are in one-to-one correspondence with universal elements of ''F''. For this reason, it is common to refer to universal elements (''A'',''u'') as representations.

			==Examples==

			* Consider the contravariant functor ''P'' : '''Set''' → '''Set''' which maps each set to its [[power set]] and each function to its [[inverse image]] map. To represent this functor we need a pair (''A'',''u'') where ''A'' is a set and ''u'' is a subset of ''A'', i.e. an element of ''P''(''A''), such that for all sets ''X'', the hom-set Hom(''X'',''A'') is isomorphic to ''P''(''X'') via Φ<sub>''X''</sub>(''f'') = (''Pf'')''u'' = ''f''<sup>–1</sup>(''u''). Take ''A'' = {0,1} and ''u'' = {1}. Given a subset ''S'' ⊆ ''X'' the corresponding function from ''X'' to ''A'' is the [[indicator function\|characteristic function]] of ''S''.
			*[[Forgetful functor]]s to '''Set''' are very often representable. In particular, a forgetful functor is represented by (''A'', ''u'') whenever ''A'' is a [[free object]] over a [[singleton set]] with generator ''u''.
			** The forgetful functor '''Grp''' → '''Set''' on the [[category of groups]] is represented by ('''Z''', 1).
			** The forgetful functor '''Ring''' → '''Set''' on the [[category of rings]] is represented by ('''Z'''[''x''], ''x''), the [[polynomial ring]] in one [[Variable (mathematics)\|variable]] with [[integer]] [[coefficient]]s.
			** The forgetful functor '''Vect''' → '''Set''' on the [[category of real vector spaces]] is represented by ('''R''', 1).
			** The forgetful functor '''Top''' → '''Set''' on the [[category of topological spaces]] is represented by any singleton topological space with its unique element.
			*A [[group (mathematics)\|group]] ''G'' can be considered a category (even a [[groupoid]]) with one object which we denote by •. A functor from ''G'' to '''Set''' then corresponds to a [[G-set\|''G''-set]]. The unique hom-functor Hom(•,–) from ''G'' to '''Set''' corresponds to the canonical ''G''-set ''G'' with the action of left multiplication. Standard arguments from group theory show that a functor from ''G'' to '''Set''' is representable if and only if the corresponding ''G''-set is simply transitive (i.e. a [[torsor\|''G''-torsor]] or [[heap (mathematics)\|heap]]). Choosing a representation amounts to choosing an identity for the heap.
			*Let ''C'' be the category of [[CW-complex]]es with morphisms given by homotopy classes of continuous functions. For each natural number ''n'' there is a contravariant functor ''H''<sup>''n''</sup> : ''C'' → '''Ab''' which assigns each CW-complex its ''n''<sup>th</sup> [[cohomology group]] (with integer coefficients). Composing this with the [[forgetful functor]] we have a contravariant functor from ''C'' to '''Set'''. [[Brown's representability theorem]] in algebraic topology says that this functor is represented by a CW-complex ''K''('''Z''',''n'') called an [[Eilenberg–Mac Lane space]].

			==Properties==
			===Uniqueness===

			Representations of functors are unique up to a unique isomorphism. That is, if (''A''<sub>1</sub>,Φ<sub>1</sub>) and (''A''<sub>2</sub>,Φ<sub>2</sub>) represent the same functor, then there exists a unique isomorphism φ : ''A''<sub>1</sub> → ''A''<sub>2</sub> such that
			:<math>\Phi_1^{-1}\circ\Phi_2 = \mathrm{Hom}(\varphi,-)</math>
			as natural isomorphisms from Hom(''A''<sub>2</sub>,–) to Hom(''A''<sub>1</sub>,–). This fact follows easily from [[Yoneda's lemma]].

			Stated in terms of universal elements: if (''A''<sub>1</sub>,''u''<sub>1</sub>) and (''A''<sub>2</sub>,''u''<sub>2</sub>) represent the same functor, then there exists a unique isomorphism φ : ''A''<sub>1</sub> → ''A''<sub>2</sub> such that
			:<math>(F\varphi)u_1 = u_2.</math>

			===Preservation of limits===

			Representable functors are naturally isomorphic to Hom functors and therefore share their properties. In particular, (covariant) representable functors [[Limit (category theory)#Preservation of limits\|preserve all limits]]. It follows that any functor which fails to preserve some limit is not representable.

			Contravariant representable functors take colimits to limits.

			===Left adjoint===

			Any functor ''K'' : ''C'' → '''Set''' with a [[left adjoint]] ''F'' : '''Set''' → ''C'' is represented by (''FX'', η<sub>''X''</sub>(•)) where ''X'' = {•} is a [[singleton set]] and η is the unit of the adjunction.

			Conversely, if ''K'' is represented by a pair (''A'', ''u'') and all small [[copower]]s of ''A'' exist in ''C'' then ''K'' has a left adjoint ''F'' which sends each set ''I'' to the ''I''th copower of ''A''.

			Therefore, if ''C'' is a category with all small copowers, a functor ''K'' : ''C'' → '''Set''' is representable if and only if it has a left adjoint.

			==Relation to universal morphisms and adjoints==

			The categorical notions of [[universal morphism]]s and [[adjoint functor]]s can both be expressed using representable functors.

			Let ''G'' : ''D'' → ''C'' be a functor and let ''X'' be an object of ''C''. Then (''A'',φ) is a universal morphism from ''X'' to ''G'' [[if and only if]] (''A'',φ) is a representation of the functor Hom<sub>''C''</sub>(''X'',''G''–) from ''D'' to '''Set'''. It follows that ''G'' has a left-adjoint ''F'' if and only if Hom<sub>''C''</sub>(''X'',''G''–) is representable for all ''X'' in ''C''. The natural isomorphism Φ<sub>''X''</sub> : Hom<sub>''D''</sub>(''FX'',–) → Hom<sub>''C''</sub>(''X'',''G''–) yields the adjointness; that is
			:<math>\Phi_{X,Y}\colon \mathrm{Hom}_{\mathcal D}(FX,Y) \to \mathrm{Hom}_{\mathcal C}(X,GY)</math>
			is a bijection for all ''X'' and ''Y''.

			The dual statements are also true. Let ''F'' : ''C'' → ''D'' be a functor and let ''Y'' be an object of ''D''. Then (''A'',φ) is a universal morphism from ''F'' to ''Y'' if and only if (''A'',φ) is a representation of the functor Hom<sub>''D''</sub>(''F''–,''Y'') from ''C'' to '''Set'''. It follows that ''F'' has a right-adjoint ''G'' if and only if Hom<sub>''D''</sub>(''F''–,''Y'') is representable for all ''Y'' in ''D''.

			== See also ==
			* [[Subobject classifier]]

			== References ==

			*{{cite book \| first = Saunders \| last = Mac Lane \| authorlink = Saunders Mac Lane \| year = 1998 \| title = [[Categories for the Working Mathematician]] \| series = Graduate Texts in Mathematics '''5''' \| edition = 2nd \| publisher = Springer \| isbn = 0-387-98403-8}}

			{{Functors}}
			[[Category:Representable functors\| ]]

en>Jackfork: Reverted edits by 93.62.230.60 (talk) to last version by 92.103.106.74

2012-08-26T09:32:05Z

Reverted edits by 93.62.230.60 (talk) to last version by 92.103.106.74

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Hi there! :) My name is Isidro, I'm a student studying Educational Studies from Fredericia, Denmark.<br><br>My homepage ... Crazy Taxi City Rush Hack ([https://Www.Facebook.com/CrazyTaxiCityRushHackToolCheatsAndroidiOS click through the up coming article])

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95.117.44.245 at 09:44, 12 December 2014

115.114.0.210: /* Commercial products */

en>HighKing: /* Open-source projects */ cleanup

en>Jackfork: Reverted edits by 93.62.230.60 (talk) to last version by 92.103.106.74