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Topic: Additive functor

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	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 (1602 words)

	Additive category: Definition and Links by Encyclopedian.com - All about Additive category
		Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subject.
		Recall that a functor F: C → D between preadditive categories is additive if it is an Abelian group homomorphism on each hom-set in C.
		In fact, it is a theorem that all adjoint functors between additive categories must be additive functors, and most interesting functors studied in all of category theory are adjoints.
	www.encyclopedian.com /ad/Additive-category.html (897 words)

	Pre-Abelian category - Wikipedia, the free encyclopedia
		Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
		First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism on each hom-set.
		Then it turns out that a functor between pre-Abelian categories is left exact if and only if it is additive and preserves all kernels, and it's right exact iff it's additive and preserves all cokernels.
	en.wikipedia.org /wiki/Pre-Abelian_category (882 words)

	Equivalence of categories
		An equivalence of categories consists of a functor between the involved categories, which is required to have an "inverse" functor.
		However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its "inverse" is not necessarily the identity mapping.
		D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category), then D may be turned into a preadditive category (or additive category, or abelian category) in such a way that F becomes an additive functor.
	pedia.newsfilter.co.uk /wikipedia/e/eq/equivalence_of_categories.html (1520 words)

	FUNCTOR FACTS AND INFORMATION
		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.
		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.
		Functors themselves can be considered as objects in a category called a functor_category.
	www.witwib.com /functor (1527 words)

	Science Fair Projects - Functor category
		In category theory, 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.all-science-fair-projects.com /science_fair_projects_encyclopedia/Category_of_functors (1113 words)

	Functor Article, Functor Information (Site not responding. Last check: 2007-10-08)
		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.
		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.
		Forgetful functors: The functor U : Grp → Set whichmaps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor.Functors like these, which "forget" some structure, are termed forgetful functors.
	www.anoca.org /functors/category/functor.html (1494 words)

	Functor category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		a standard construction embeds a given category in a functor category; the functor category has much nicer properties than the original category, allowing to perform certain operations that were not available in the original setting.
		The embedding of the category C in a functor category that was mentioned earlier uses the (Click link for more info and facts about Yoneda lemma) Yoneda lemma as its main tool.
		The category Cat of all small categories with functors as morphisms is therefore a (Click link for more info and facts about cartesian closed category) cartesian closed category.
	www.absoluteastronomy.com /encyclopedia/f/fu/functor_category.htm (1081 words)

	Functor - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-08)
		That is, instead of saying $F: C\rightarrow D$ is a contravariant functor, they simply write $F: C^{op} \rightarrow D$ (or sometimes $F:C \rightarrow D^{op}$ ) and call it a functor.
		Power sets: The power set functor P : Set → Set maps each set to its power set and each function $f : X \subseteq Y$ to the map which sends $U \subseteq X$ to its image $f(U) \subseteq Y$ .
		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.arikah.net /encyclopedia/Functor (1588 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		D be an additive functor, and denote by D0 the full subcategory of* * D formed by the objects in the image of F.
		Moreover, The* orem 4.4 implies that F is a cohomological quotient functor*.
		D be an additive functor and suppose F is surjective on objects.
	hopf.math.purdue.edu /KrauseH/quotient.txt (17394 words)

	Isomorphic categories (Site not responding. Last check: 2007-10-08)
		An equivalence of categories consists of a functor between the involvedcategories, which is required to have an "inverse" functor.
		However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its"inverse" is not necessarily the identity mapping.
		D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category),then D may be turned into a preadditive category (or additive category, or abelian category) in such a way thatF becomes an additive functor.
	www.therfcc.org /isomorphic-categories-215786.html (1075 words)

	Preadditive category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		If C and D are preadditive categories, then a (Click link for more info and facts about functor) functor F: C → D is additive if it too is (Click link for more info and facts about enriched) enriched over the category Ab.
		For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a (Click link for more info and facts about ring homomorphism) ring homomorphism from R to S is represented by an additive functor from R to S, and conversely.
		If C and D are categories and D is preadditive, then the (Click link for more info and facts about functor category) functor category Fun(C,D) is also preadditive, because (Click link for more info and facts about natural transformation) natural transformations can be added in a natural way.
	www.absoluteastronomy.com /encyclopedia/p/pr/preadditive_category.htm (1551 words)

	[No title]
		A homology theory on a triangulated category S is an exact functor to an Abelian category which preserves the coproducts that exist in S. Unless we state otherwise, the target category will always be taken to be the category Ab of Abelian groups.
		A functor in A is a homology theory if and only if it has finite projective dimension if and only if it has projective dimension at most one.
		We then define Ind (F) to be the subcategory of all functors F 2 B that can be written as a filtered colimit of a small diagram of objects of F. It is equivalent to require that the category of pairs (X, a) (where X 2 F and a 2 F X) is filtered.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	Functor (Site not responding. Last check: 2007-10-08)
		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.
		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.
		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.yotor.com /wiki/en/fu/Functor.htm (1530 words)

	Exact functor (Site not responding. Last check: 2007-10-08)
		The functor whichassociates to each sheave L the group of global sections L(X) is left-exact.
		The degree to which a left exact functor fails to be exact can be measured with its right derived functors ; the degree to which a right exact functor fails to be exact can be measured withits left derived functors.
		Left- and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint to G, then F is right exact and Gis left exact.
	www.therfcc.org /exact-functor-210917.html (417 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		C(G): Here A(G) is the Burnside ring of G and C(G) is the additive group of continu* *ous functions from the space of subgroups of G to the integers, where subgroups are understood to be closed.
		Therefore the co* m- posite functor* on C that sends an object Y to ss0(Y)R P is a cohomology functo* *r.
		Since the functors -V 1 preserve retracts, it is clear that the G-spectra of the statement are dualizable.
	hopf.math.purdue.edu /Fausk-Lewis-May/FLMApril20.txt (5456 words)

	ipedia.com: Functor Article (Site not responding. Last check: 2007-10-08)
		Functors can be thought of as morphisms in the catego...
		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 its image f(U) ⊆ Y.
		The Yoneda lemma explains that often a category C can be extended by considering a category of pre-sheaves on C.
	www.ipedia.com /functor.html (1514 words)

	math lessons - Isomorphism of categories
		In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e.
		A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets.
		The functor category of all additive functors from this category to the category of abelian groups is isomorphic to the category of left modules over the ring.
	www.mathdaily.com /lessons/Isomorphism_of_categories (623 words)

	Tor functor -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		As is true for every family of derived functors, every (Click link for more info and facts about short exact sequence) short exact sequence
		The reason: every abelian group A has a free resolution of length 2, since subgroups of (Click link for more info and facts about free abelian group) free abelian groups are free abelian.
		The Tor functors commute with arbitrary (A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets) direct sums: there is a (Click link for more info and facts about natural isomorphism) natural isomorphism
	www.absoluteastronomy.com /encyclopedia/T/To/Tor_functor.htm (425 words)

	Module (mathematics) - Wikipedia, the free encyclopedia
		Any ring R can be viewed as a preadditive category with a single object.
		With this understanding, a left R-module is nothing but a (covariant) additive functor from R to the category Ab of abelian groups.
		This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab should be considered a generalized left module over C; these functors form a functor category C-Mod which is the natural generalization of the module category R-Mod.
	www.wikipedia.org /wiki/Submodule (1345 words)

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