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| 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.
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| 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]].
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| ==Definition==
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| 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'').
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| 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
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| :Φ : Hom(''A'',–) → ''F''
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| is a natural isomorphism.
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| 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'''.
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| ==Universal elements==
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| 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
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| :<math>u = \Phi_A(\mathrm{id}_A).\,</math>
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| Conversely, given any element ''u'' ∈ ''F''(''A'') we may define a natural transformation Φ : Hom(''A'',–) → ''F'' via
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| :<math>\Phi_X(f) = (Ff)(u)\,</math>
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| 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:
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| :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''.
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| 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''.
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| 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.
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| ==Examples==
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| * 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''.
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| *[[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''.
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| ** The forgetful functor '''Grp''' → '''Set''' on the [[category of groups]] is represented by ('''Z''', 1).
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| ** 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.
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| ** The forgetful functor '''Vect''' → '''Set''' on the [[category of real vector spaces]] is represented by ('''R''', 1).
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| ** The forgetful functor '''Top''' → '''Set''' on the [[category of topological spaces]] is represented by any singleton topological space with its unique element.
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| *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.
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| *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]].
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| ==Properties==
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| ===Uniqueness===
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| 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
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| :<math>\Phi_1^{-1}\circ\Phi_2 = \mathrm{Hom}(\varphi,-)</math>
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| as natural isomorphisms from Hom(''A''<sub>2</sub>,–) to Hom(''A''<sub>1</sub>,–). This fact follows easily from [[Yoneda's lemma]].
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| 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
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| :<math>(F\varphi)u_1 = u_2.</math>
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| ===Preservation of limits===
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| 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.
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| Contravariant representable functors take colimits to limits.
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| ===Left adjoint===
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| 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.
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| 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''.
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| 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.
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| ==Relation to universal morphisms and adjoints==
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| The categorical notions of [[universal morphism]]s and [[adjoint functor]]s can both be expressed using representable functors.
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| 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
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| :<math>\Phi_{X,Y}\colon \mathrm{Hom}_{\mathcal D}(FX,Y) \to \mathrm{Hom}_{\mathcal C}(X,GY)</math>
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| is a bijection for all ''X'' and ''Y''.
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| 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''.
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| == See also ==
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| * [[Subobject classifier]]
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| == References ==
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| *{{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}}
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| {{Functors}}
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| [[Category:Representable functors| ]]
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Games have become like movies when it comes to versatility. They have people come out and try to make a goal from center ice. Why not throw in other types of exercise along with riding. This will give you a nice team of a Coder, Writer, Graphic Designer, and Music Composer. MAG gives off the feeling, probably because its true, that there is a large scale conflict going on, everywhere nearby. should you are calm, then you definitely commence from your fine basis to begin the recovery.
The levels are full of secret areas which hold power-ups to reward players. In order to change the game from Tablero to Toblaro, the coins are replace with shot glasses filled with alcoholic beverages, when the money (or the glasses) are captured, the winner gets to drink it and the loser must refill the contents with his alcohol.
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