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	Derived functor - Wikipedia, the free encyclopedia
		The functor which assigns to each such sheaf L the group L(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H
		Derived functors and the long exact sequences are "natural" in several technical senses.
		becomes a functor from the functor category of all left exact functors from A to B to the full functor category of all functors from A to B.
	en.wikipedia.org /wiki/Derived_functor (1179 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.
		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.
		derived functor the image of a short exact sequence under a functor that is only half-exact can be extended to a long exact sequence.
	en.wikipedia.org /wiki/Functor (1602 words)

	Functor -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		Functors can be thought of as (Click link for more info and facts about morphism) morphisms in the category of all ((The slender part of the back) small) categories.
		A functor from G to Set is nothing but a (Action taken by a group of people) group action of G on a particular set, i.e.
		Another example is the functor Rng → Ab which maps a (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) ring to its underlying additive (A group that satisfies the commutative law) abelian group.
	www.absoluteastronomy.com /encyclopedia/f/fu/functor.htm (1718 words)

	On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology, by Guillermo Cortiñas (Site not responding. Last check: 2007-11-06)
		On the derived functor analogy in the Cuntz-Quillen framework for cyclic homology, by Guillermo Cortiñas
		The purpose of this paper is to formalize this derived functor analogy.
		We show that the localization of the category of countably indexed pro-algebras at the class of deformations exists, and that periodic cyclic homology is the derived functor of de Rham (co)homology with respect to this localization.
	www.math.uiuc.edu /K-theory/0223 (148 words)

	PlanetMath: derived functor
		Note that a completely analogous construction can be done for right-exact functors and for contravariant functors exact on either side, but it is traditional to only describe one case, as doing the others mostly consists of reversing arrows (and replacing ``injective'' with projective when appropriate).
		Sheaf cohomology arises as the right derived functors of the global section functor on sheaves.
		This is version 13 of derived functor, born on 2003-02-10, modified 2004-12-31.
	planetmath.org /encyclopedia/DerivedFunctor.html (375 words)

	Functor Article, Functor Information (Site not responding. Last check: 2007-11-06)
		Functors were first considered in algebraic topology, wherealgebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
		Constant functor: A very boring functor C → D is one which maps every object ofC to a fixed object X in D and every morphism in C to the identity morphism on X.Such a functor is called a constant or selection functor.
		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 algebraC(X) of all real-valued continuous functions on that space.
	www.anoca.org /functors/category/functor.html (1494 words)

	MoreRead.com - .NET style delegates for VC++ 6 - 2003-08-19 (Site not responding. Last check: 2007-11-06)
		Functors become much more useful when you want to call functions that look the same (have the same parameters and return value) without knowing what class or object they belong to.
		So functors can be pretty handy things, but it's a bit of a pain to have to rewrite the classes if different numbers of parameters or a different return type are required.
		The delegate and functor classes are put in their own namespace, so that they are one manageable unit.
	www.moreread.com /it/archives/2003/08/19/29896.shtml (2014 words)

	[No title]
		The functor C_from G-spaces to OG -algebras is the composite of the functor* : GTop -!
		OG E. Since all of these functors preserve all weak equivalences, the composite of the deri* ved functors* is the derived functor of the composite.
		The category of contravar* i- ant functors* from D to simplicial sets is a closed model category with weak equ* iva- lences the objectwise weak equivalences and fibrations the objectwise Kan fibra *tions.
	hopf.math.purdue.edu /Mandell/finite.txt (5069 words)

	Functor (Site not responding. Last check: 2007-11-06)
		In category theory, a functor is a mapping from one category to another which maps objects toobjects and morphisms to morphisms in such a manner that the composition of morphisms and the identities are preserved.
		Functors were first considered in algebraic topology, wherealgebraic objects are associated to topological spaces, andalgebraic homomorphisms are associated to continuous maps.
		Nowadays, functors are used throughout modernmathematics to relate various categories.
	www.therfcc.org /functor-35866.html (106 words)

	SML/NJ Special Features (Site not responding. Last check: 2007-11-06)
		But there are occasions when one like to parameterize over functors as well as structures, which requires a truly higher-order module system (see, for instance, the powerset functor example.
		Functor specifications were already part of the module syntax of the Definition of Standard ML (Figure 8, p.
		A common use of functors returning functors in their result is to approximate a curried functor with multiple parameters.
	www.smlnj.org /doc/features.html (1135 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		An important corollary of the fact that a derived category D(A) is the homotopy category of a model category is that the group D(A)(X, Y) of maps is a set (as opposed to a proper class) for any two chain complexes X and Y.
		P Ext (-, B)) for the derived* functors of A(-, B) with respect to the categorical (resp.
		In addition to the connection between phantom maps and pure homological algebra, the authors are interested in the pure derived category as a tool for connecting the global pure dimension of a ring R to the behaviour of phantom maps in DC and DP under composition.
	jdc.math.uwo.ca /papers/relative.txt (10317 words)

	Callbacks in C++ using template functors
		These Functor classes are sufficient to meet the callback needs of component designers, as they offer a standard and consistent way to offer callback services, and a simple mechanism for invoking the callback function.
		The Functor class is parameterized on the types of the callback function signature, holds the callee data in a typeless manner, and defines a typed operators but doesn't actually perform the work of calling back.
		Here, makefunctor is parameterized with the type of the argument to the Functor, the type of the Callee, the type of the class of which the member-function is a member, and the argument and return types of the member function.
	www.contactor.se /~daniel/links/callback.html (5733 words)

	ipedia.com: Functor Article (Site not responding. Last check: 2007-11-06)
		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.
		Universal constructions: Functors are often defined by universal properties; examples are the tensor product discussed above, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
	www.ipedia.com /functor.html (1514 words)

	Derived Categories for Dummies, Part IV \| The String Coffee Table
		For instance this derived categroy of coherent sheaves is equivalent to what is called a triangulated Fukaya categroy and also to (at least for a large number of cases) the derived category of representations of some quiver (which I mentioned already in part III).
		The derivation of this could be summarized by a couple of sub-steps, which are however not strictly necessary in order to move on to step 4.
		These complexes are of course precisely the objects in the derived category of coherent sheaves, as described in part I.
	golem.ph.utexas.edu /string/archives/000538.html (1326 words)

	[No title]
		D be an additive functor, and denote by D0 the full subcategory of* * D formed by the objects in the image of F.
		We conclude that F is not a cohomological quotient funct* *or.
		Moreover, The* orem 4.4 implies that F is a cohomological quotient functor*.
	hopf.math.purdue.edu /KrauseH/quotient.txt (17394 words)

	Alice Manual - Extensions to the Module Language
		Derived forms analogous to the core language are provided for defining functors that should evaluate lazily or in a separate threads:
		The derived form for functor arguments, allowing a list dec of declarations being given instead of a structure expression, has been generalized: in a structure expression, parentheses may either enclose another strexp, or a dec.
		In this example, the functor application is performed solely for its side effect, and does not return any interesting result.
	www.ps.uni-sb.de /alice/manual/modules.html (1196 words)

	Samuel Eilenberg, September 30, 1913—January 30, 1998 \| By Hyman Bass, Henri Cartan, Peter Freyd, Alex Heller, and ...
		In each of these cases the cohomology groups in question were the derived functors of naturally occurring Hom functors.
		All these various examples of the construction of new functors as "derived" functors of given ones were at hand for Eilenberg.
		Categories were defined in order to define functors, which in turn were defined in order to define natural transformations, which were defined finally in order to prove theorems that could not be proved before.
	stills.nap.edu /html/biomems/seilenberg.html (6599 words)

	Functor
		In category theory, a functor is a mapping from one category to another which maps objects to objects and morphisms to morphisms in such a manner that the composition of morphisms and the identities are preserved.
		Functors were first considered in algebraic topology, where algebraic objects are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps.
		For the precise definition and examples, see the article on category theory.
	www.brainyencyclopedia.com /encyclopedia/f/fu/functor.html (162 words)

	Derived Categories for the Working Mathematician - Thomas (ResearchIndex)
		It is becoming increasingly dicult for geometers and even physicists to avoid papers containing phrases like \triangulated category", not to mention derived functors.
		I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology.
		for an introduction to the derived category or [49] for more detail) The inverse functor is easily determined because there is an...
	citeseer.ist.psu.edu /thomas00derived.html (297 words)

	Derived Categories for Dummies, Part II \| The String Coffee Table
		I spent yesterday sitting on the beach at Vietri and reading Weibel, ‘An introduction to homological algebra’, trying to understand the details of derived functors.
		are called the hyper-derived functors, by definition (5.7.4).
		In order to understand hyper-derived functors it is finally necessary to first understand ordinary derived functors (to be distinguished from the ‘total’ derived functors that I started with).
	golem.ph.utexas.edu /string/archives/000535.html (493 words)

	Some Remarks on the Constants Functor (ResearchIndex)
		In this paper we study the cohomological properties of the constant functor on di erential modules.
		Since this functor is left exact, it has a right derived functor, which turns out to be related to the Koszul cohomology over the ring of linear differential operators.
		To study vanishing of the cohomology groups (acyclicity), we introduce the condition of having partial antiderivatives, which is a weaker condition than being...
	citeseer.ist.psu.edu /kovacic01some.html (261 words)

	Functor (Site not responding. Last check: 2007-11-06)
		Power sets: The power set functor P : Set → Set maps each set to its power set and each function f : X → Y to the map which sends U ⊆ X to itsf(U) ⊆ Y.
		In itself, this category is not very exciting, but the functors from C
		This motivating example is generalized by considering pre-sheaves on arbitrary categories: a pre-sheaf on C is a functor defined on C
	www.yotor.com /wiki/en/fu/Functor.htm (1530 words)

	Info Node: (sml)CL4 (Site not responding. Last check: 2007-11-06)
		It was felt to be justified for the following reasons; 1) It would be intolerable to have two equally heavy sets of rules for the two styles of using functors; a way had to be found to express one in terms of the other.
		Suppose now that sigexp is SIG LOCAL OPEN A IN spec END END Then only the structure and type constructor components of A are relevant, since a specification spec cannot refer to external value and exception identifiers.
		This includes its use in deriving the derived functor forms.
	www.math.psu.edu /bin/info2www?(sml)CL4 (340 words)

	TQFT Club Meeting 6-May-1999 (Site not responding. Last check: 2007-11-06)
		The derived category is the algebraic analog of the homotopy category of topological spaces.
		I will not discuss this analogy, but rather content myself with developing the notion of total derived functor.
		The language of derived categories is the appropriate and natural one for understanding derived functors, and its jargon is becoming increasingly abundant in the hep/math literature.
	www.math.ist.utl.pt /~rpicken/tqft/06_05_99.html (374 words)

	derived - OneLook Dictionary Search
		Derived : Online Plain Text English Dictionary [home, info]
		Example: "The belief that classes and organizations are secondary and derived- John Dewey"
		Phrases that include derived: derived function, platelet derived growth factor, derived demand, derived functor, derived protein, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=derived (144 words)

	functor - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "functor" is defined.
		Functor : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include functor: contravariant functor, covariant functor, derived functor, exact functor, adjoint functor, more...
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=functor (145 words)

	[No title]
		In fact this is probably the very first example of a derived functor that anyone ever thought of.
		At some point, someone realized that you could systematize all this stuff using the concept of "derived functor".
		If by "G-module" you mean an abelian group with an action of the group G, then G-modules do form an abelian category, and you can play the derived functor game.
	www.math.niu.edu /~rusin/known-math/00_incoming/postnikov (1497 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The class is a predefined functor for the Gaussian bell function.
		The parameters reflect the position (the d first values) and the width (the next d values, where d is the number of space dimen sions); It is convenient to read the parameters and the functor name using FieldFormat (the data member function_prm can be fed directly into the this functor).
		There are two ways to initialize a BellFunc object: 1) call Bell Func(prms), where prms is a vector of parameters, or 2) first declare an empty object using the constructor without arguments and then call setParameters(prms).
	www.diffpack.com /diffpack/refmanuals/dp30/BellFunc.html (174 words)

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