
	Where results make sense

Topic: Exact functor

Related Topics

Derived functor

Functor

List of category theory topics

Injective module

Projective module

Abelian group

Harry Nilsson

Welfen

In the News (Wed 11 Nov 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	Exact functor - Wikipedia, the free encyclopedia
		In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences.
		Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily.
		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 with its left derived functors.
	en.wikipedia.org /wiki/Exact_functor (644 words)

	PlanetMath: derived functor
		A completely analogous construction can be carried out 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), and the result is that of a left derived functor
		which is natural (a morphism of short exact sequences induces a morphism of long exact sequences).
		This is version 17 of derived functor, born on 2003-02-10, modified 2006-05-15.
	planetmath.org /encyclopedia/DerivedFunctor.html (382 words)

	[No title]
		Mod D is exact because F is flat, and it is a qu* otient functor* because F* is fully faithful.
		Moreover, The* orem 4.4 implies that F is a cohomological quotient functor*.
		Thus I is exact because Fc is* * a cohomological quotient functor by Theorem 11.1.
	www.math.purdue.edu /research/atopology/KrauseH/quotient.txt (17394 words)

	[No title]
		Exactness of f0implies that f is cohom* *ological, since h is cohomological.
		The functor g induces an equivalence between C=B and B? Proof.An inverse is the composition of the inclusion B? C with the quotient f* *unctor_ C !
		D is any exact functor into a triangulated category D and C is any object in C0) is isomorphic to Hom (-; X)C0 for some object X in C. This condition is a weak form of Brown representability.
	www.math.purdue.edu /research/atopology/KrauseH/smash.txt (13290 words)

	Functor - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-03)
		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.cob-web.org:8888 /wiki/Functor (1848 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)

	Derived functor - Wikipedia, the free encyclopedia
		Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B.
		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
		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 (1185 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.
		We suppose therefore that the short exact sequence of functors is given, and that F is the restriction of a representable.
		The exact sequence of functors is not split, and we __ conclude that F cannot be isomorphic to any yY.
	jdc.math.uwo.ca /papers/purity.txt (7106 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		It is functorial in the sense that exact functors give rise to `stable functors'.
		Triangle functors, equivalences and adjoints are presented in Section 8.
		Derived functors are constructed using a `generalized calculus of fractions'.
	www1.elsevier.com /homepage/saj/523281/h19.htm (346 words)

	Derived Functors (Site not responding. Last check: 2007-11-03)
		Of course, it is not obvious that such sequence of functors exist.
		First observe that to prove the existence of satellite functors in all degrees, it is enough to prove existence of
		Consider the following diagram where first row is exact and second is a complex.
	www.imsc.res.in /~sgautam/main/node7.html (409 words)

	Noncommutative topology - homotopy functors and E-theory (Site not responding. Last check: 2007-11-03)
		In order to understand the role of the E bifunctor we are lead to examine the general properties of homotopy functors on the category of C*-algebras.
		We examine cofibrations in the category of C-algebras, and introduce the notion of a `cofibration half exact' functor* as being the natural generalization of the topological concept of an exact functor.
		An excisive functor on the category of C-algebras is defined to be one which does not distinguish between the kernel and the `homotopy kernel' (the mapping cone) of a surjective -homomorphism.
	www-math.mit.edu /~jg/papers/etheory.html (284 words)

	PlanetMath: exact functor
		is said to be right exact if whenever
		A (covariant or contravariant) functor is said to be exact if it is both left exact and right exact.
		This is version 3 of exact functor, born on 2002-01-05, modified 2003-09-20.
	planetmath.org /encyclopedia/ExactFunctor.html (87 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		The notion of a triangle functor (=$S$-functor \cite{kv} =exact functor \cite{del}) appears as the natural axiomatization of this concept.
		Triangle functors, equivalences and adjoints are presented in section \ref{trian-fun}.
		In section \ref{der-cat-ex-subcat}, we give a sufficient condition for an inclusion of exact categories to induce an equivalence of their derived categories.
	www.math.jussieu.fr /~keller/publ/dcuabs.txt (342 words)

	Exact Functor -- from Wolfram MathWorld
		A functor between categories of groups or modules is called exact if it preserves the exactness of sequences, or equivalently, if it transforms
		A covariant functor is called left exact if it preserves the exactness of all sequences
		iff it is both left and right exact.
	mathworld.wolfram.com /ExactFunctor.html (110 words)

	Springer Online Reference Works
		is a left-exact functor from the category of sheaves of Abelian groups on
		Its derived functors are the local cohomology functors
		These cohomology functors can be explicitly calculated using Koszul complexes, cf.
	eom.springer.de /L/l060090.htm (348 words)

	Subject Index
		-proper short exact sequence in a category of complexes 617
		projectivity with respect to a class of short exact sequences 616
		relative cohomology of a functor with coefficients in an abelian group valued functor 625
	www1.elsevier.com /homepage/saj/523281/subject.htm (1223 words)

	exact functor - OneLook Dictionary Search
		Tip: Click on the first link on a line below to go directly to a page where "exact functor" is defined.
		Exact Functor : Eric Weisstein's World of Mathematics [home, info]
		Phrases that include exact functor: left exact functor
	www.onelook.com /?w=exact+functor (89 words)

	Math 55a: A preview of ``abstract nonsense'' (Site not responding. Last check: 2007-11-03)
		must be the zero map.] The sequence is said to be ``exact'' if it is exact at each vector space with an incoming and outgoing arrow.
		An exact sequence 0 --> U --> V --> V/U --> 0 is called a ``short exact sequence''; an exact sequence involving four or more vector spaces between the initial and final zero is called a ``long exact sequence''.
		In any exact sequence of finite-dimensional vector spaces with an initial and final zero, the dimensions of the even- and odd-numbered vector spaces in the sequence have the same sum; in other words, the alternating sum of the dimensions (a.k.a.
	www.math.harvard.edu /~elkies/M55a.05/nonsense.html (424 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)

	[No title] (Site not responding. Last check: 2007-11-03)
		> I have to show that every left exact covariant functor T : R-Mod ---> Ab > preserves pullbacks, where R-Mod is the category of left R-modules.
		It may be helpful to get hold of a basic book on categories.
		direct sums and kernals (this is why left exact is relevant), you are
	mathforum.org /kb/plaintext.jspa?messageID=3689689 (250 words)

	Hexapedia - Exact functor (via CobWeb/3.1 planetlab2.netlab.uky.edu) (Site not responding. Last check: 2007-11-03)
		In category theory and its applications in mathematics, a covariant additive functor between abelian categories is called
		exact if it is left exact and right exact, i.e.
		To check whether an additive functor is exact, it is enough to check that it transforms short exact sequences into short exact sequence.
	www.hexafind.com.cob-web.org:8888 /encyclopedia/Exact_functor (488 words)

Try your search on: Qwika (all wikis)