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	Representable functor - Wikipedia, the free encyclopedia
		In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets.
		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.
		We therefore conclude that representations of F are in one-to-one correspondence with universal elements of F.
	en.wikipedia.org /wiki/Representable_functor (992 words)

	Functor - Wikipedia, the free encyclopedia
		Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
		Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space.
		Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
	en.wikipedia.org /wiki/Functor (1790 words)

	Subfunctor - Wikipedia, the free encyclopedia
		In category theory, a branch of mathematics, a subfunctor is a special type of functor which is an analogue of a subset.
		Subfunctors are also used in the construction of representable functors on the category of ringed spaces.
		Hom(−, X) is a representable functor Hom(−, Y), and that the morphism Y→X defined by the Yoneda lemma is an open immersions.
	en.wikipedia.org /wiki/Subfunctor (452 words)

	Functor category - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-20)
		In category theory, a branch of mathematics, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors.
		The category of presheaves on a topological space X is a functor category: we turn the topological space in a category C having the open sets in X as objects and a single morphism from U to V iff U is contained in V.
		The embedding of the category C in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool.
	www.sciencedaily.com /encyclopedia/functor_category (989 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		The functor of taking a Cartesian power is also representable: a representing object is a set whose cardinality equals the given power.
		A representable functor is an analogue of the concept of a "free universal algebra with one generator" .
		The theorem (above) characterizing natural transformations from a representable functor to an arbitrary functor is commonly called the Yoneda lemma.
	eom.springer.de /R/r081340.htm (401 words)

	Category theory
		Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second.
		Dual vectorspace: an example of a contravariant functor from the category of all real vector spaces to the category of all real vector spaces is given by assigning to every vector space its dual space and to every linear map its dual or transpose.
		Forgetful functors: the functor F : Ring -> Ab which maps a ring to its underlying abelian additive group.
	www.ebroadcast.com.au /lookup/encyclopedia/ca/Category_theory.html (2075 words)

	K-theory and analysis
		Examples of representable functors are ordinary cohomology (here Y is an `Eilenberg-MacLane’ space) and K-theory (where the representing spaces Y are given by the unitary groups and their classifying spaces), though there are many others of great interest.
		Similarly, if the representable functor takes values in, say, rings, Y will be a ring-space and its homology will be a ring object in the category of coalgebras, an object called by some a Hopf ring, or (better when you start to deal with these in the necessary generality) a coalgebraic ring.
		There are many cases where related functors have clearly had relationships between their representing spaces, but this is only to be seen by ad-hoc calculation; my aim has been to develop the language to enable explanations at a structural level to be approached.
	www.math.le.ac.uk /people/jhunton/coal.html (658 words)

	Yoneda lemma (Site not responding. Last check: 2007-10-20)
		Generally speaking, the Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (where Set is the category of all sets with functions as morphisms).
		This approach is akin to (and in fact generalizes) the common method of studying a ring by investigating the modules over that ring.
		The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category.
	bopedia.com /en/wikipedia/y/yo/yoneda_lemma.html (661 words)

	Category theory - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-20)
		By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them, we are studying the relationships between various classes of mathematical structure.
		A contravariant functor F from C to D is a functor that "turns morphisms around" ("reverses all the arrows").
		Adjoint functors: A functor can be left (or right) adjoint to another functor that maps in the opposite direction.
	www.sciencedaily.com /encyclopedia/category_theory (2396 words)

	Upto11.net - Wikipedia Article for Functor (Site not responding. Last check: 2007-10-20)
		Constant functor: A very boring functor C andrarr; D is one which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X.
		Forgetful functors: The functor U : Grp andrarr; Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.
		Another example is the functor Rng andrarr; Ab which maps a ring to its underlying additive abelian group.
	www.upto11.net /generic_wiki.php?q=functor (1425 words)

	PlanetMath: representable functor
		A vast number of important objects in mathematics are defined as representing functors.
		This should at least in part be viewed as the motivation for determining representability.
		This is version 8 of representable functor, born on 2001-12-12, modified 2005-12-21.
	planetmath.org /encyclopedia/RepresentableFunctor.html (218 words)

	Category theory (Site not responding. Last check: 2007-10-20)
		Then it becomes possible to relate categories by functors generalizations of functions which associate to every object of category an object of another category and every morphism in the first category a in the second.
		Algebra of continuous functions: a contravariant functor from the category topological spaces (with continuous maps as morphisms) to category of real associative algebras is given by assigning to every space X the algebra C(X) of all real-valued continuous functions on space.
		Functors like these are called representable and a major goal in many is to determine whether a given functor representable.
	www.freeglossary.com /Category_theory (3196 words)

	Representable functor - TheBestLinks.com - Bijection, Category theory, Functor, Homomorphism, ... (Site not responding. Last check: 2007-10-20)
		An arbitrary functor $F:\mathcal C\rightarrow\mathbf{Set} is said to be 'represented by a pair', (A,\phi), where A is an object of \mathcal C and \phi is in F(A), if there is a natural isomorphism \Phi:\mathrm{Hom}_{\mathcal C}(A,\;\cdot\;)\rightarrow F, given by the consistent family of bijections \Phi_X:\mathrm{Hom}_{\mathcal C}(A,X)\rightarrow F(X), such that$
		Proposition: A functor, $G:\mathcal C\rightarrow\mathcal D, has a left adjoint if and only if, for every A in \mathcal C, the$ functor from $\mathcal D to \mathrm{Set} mapping B to \mathrm{Hom}_{\mathcal C}(A,G(B)) is$ representable.
		If $(F(A),\phi) represents this$ functor then $F is the object part of a left-adjoint of G for which the isomorphism \Phi_B is functorial in B and yields the adjointness.$
	www.thebestlinks.com /Representable_functor.html (719 words)

	Brown's representability theorem - Wikipedia, the free encyclopedia
		In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions on a contravariant functor F on the homotopy category Hot of pointed CW complexes, to the category of sets Set, to be a representable functor.
		According to a combinatorial result in category theory, all finite colimits are built up from coproducts and pushouts (or just coequalisers).
		The statement of Brown's representability theorem is then that F is a representable functor on Hot (up to equivalence of functors) if and only if the wedge axiom and Mayer-Vietoris axiom are satisfied by F.
	en.wikipedia.org /wiki/Brown%27s_representability_theorem (267 words)

	Moduli of Curves (Graduate Texts in Mathematics) by Joe Harris [ISBN: 0387984380] - Find Cheap Textbook Prices & Save ...
		Putting an equivalence relation on the families gives a functor, called the moduli functor, which acts on the category of schemes to the category of sets.
		The functor is representable in the category of schemes if there is an isomorphism between the functor and the functor of points of a scheme.
		This particular scheme is called the fine moduli space for the functor, as distinguished from the coarse moduli space, where the functor is not representable, i.e.
	www.gettextbooks.com /isbn_0387984380.html (999 words)

	Natural transformation
		Natural transformations are, after categories and functors, one of the most basic notions of categorical algebra and consequently appear in the majority of its applications.
		Two functors F and G are called naturally isomorphic if there exists a natural isomorphism from F to G.
		Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly, and those of the singular theory, would be isomorphic.
	www.teachtime.com /en/wikipedia/n/na/natural_transformation.html (529 words)

	Representable functor: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-10-20)
		A forgetful functor is a type of functor in mathematics....
		The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory....
		If represents this functor then is the object part of a left-adjoint of for which the isomorphism is functorial in and yields the adjointness.
	www.absoluteastronomy.com /encyclopedia/r/re/representable_functor.htm (1512 words)

	Yoneda lemma (Site not responding. Last check: 2007-10-20)
		It also how the embedded category of representable functors their natural transformations relates to the other objects in larger functor category.
		This functor is called the Yoneda embedding and it is "natural" in the that every functor C → D induces a commutative diagram
		The Yoneda lemma remains true for preadditive if we choose as our extension the of additive contravariant functors from the original category the category of abelian groups; these are which are compatible with the addition of and should be thought of as forming module category over the original category.
	www.freeglossary.com /Yoneda_lemma (763 words)

	Lambek and Scott: Introduction to higher order categorical logic (Site not responding. Last check: 2007-10-20)
		Equivalently, an equivalence is a pair of functors F:A->B and U:B->A with FU iso to 1_B and UF iso to 1_A.
		A functor U:B->A is called "tripleable" or "monadic" if it is a right adjoint and if the comparison functor is an equivalence of categories (so that up to equivalence B is the Eilenberg-Moore category).
		K is a functor since K(eta_A) = eps'_F'A o F'(eta_A) = 1_A, and K(g*f) = eps'F'A'' o F'(mu_A'' o T(g) o f) = eps'F'A'' o F'U'eps'F'A'' o F'U'F'g o F'f = eps'F'A'' o eps'F'U'F'A'' o F'U'F'g o F'f = eps'F'A'' o F'g o eps'F'A' o F'f = K(g) o K(f).
	www.andrew.cmu.edu /user/cebrown/notes/lambekscott.html (4587 words)

	[No title]
		Ab are the restrictions of representable functors on T, and all natural transformations are the restrictions of morphisms in T. It has been something of a mystery, to what extent this generalises to other trian- gulated categories.
		Failure of Brown representability 13 References 22 Introduction The introduction is written for the reader who knows about derived categories, but is not necessarily familiar with previous articles by the authors and their close friends.
		We suppose therefore that the short exact sequence of functors is given, and that F is the restriction of a representable.
	jdc.math.uwo.ca /papers/purity.txt (7106 words)

	Wikinfo \| Functor (Site not responding. Last check: 2007-10-20)
		F(g ο f) = F(g) ο F(f) for all morphisms f:X → Y and g:Y→ Z. That is, functors must preserve identity morphisms and composition of morphism.
		Dual vectorspace: The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself.
		Forgetful functors: The functor U : Grp → Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.
	www.wikinfo.org /wiki.php?title=Functor (1552 words)

	[No title]
		It is shown that Brown representability holds for a compactly gen* erated triangulated category if and only if for every additive functor* from the category of c* ompact objects into the category of abelian groups a flat cover can be constructed in a canon *ical way.
		The proof also shows that Brown representability for objects and morphisms is a con* sequence of Brown representability* for objects and isomorphisms.
		The proof also sh* ows that Brown representability* for objects and morphisms is a consequence of Brown repr* *esentability for objects and isomorphisms.
	www.math.purdue.edu /research/atopology/KrauseH/brown.txt (2147 words)

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