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Topic: Left exact functor

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exact sequence

exact functor

short exact sequence

Closed and exact differential forms

Signing Exact English

Exact trigonometric constants

long exact sequence

left exact

left exact functor

Exact coloring

Exact inexact arithmetic

Exact count

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	Exact functor - Wikipedia, the free encyclopedia
		In homological algebra, an exact functor is one which preserves exact sequences.
		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.
		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 G is left exact.
	en.wikipedia.org /wiki/Exact_functor (597 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)

	Encyclopedia :: encyclopedia : Category theory (Site not responding. Last check: 2007-11-01)
		Such a process is called a functor, and it associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second.
		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.hallencyclopedia.com /Category_theory (2343 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]
		This notion exploits a strong interplay between representations of the general linear group and the symmetric group coming from the action of these groups on the tensor power of a space, and may be thought of as a generalization of a classical notion of symmetrization of a representation.
		The exactness of evaluation on an arbi- trary Y -permutative module follows from the fact, that any such a module is a direct summand in a finite sum of V d for a space V of dimension d.
		Namely, our formula says that the functor F and the d-module S(F) are obtained by applying the same d-functor to the right d-structure of: V d in the first case, and k[ d] in the second.
	www.math.purdue.edu /research/atopology/Chalupnik/extpol.txt (7587 words)

	PlanetMath: exact functor
		is said to be left 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)

	Everything about East (Site not responding. Last check: 2007-11-01)
		On the contrary, the left side of a star chart (a map of the skies) is east.
		By convention, the left side of a map is west.
		The extent to which justice is conventional (as opposed to natural or objective) is historically an important debate among philosophers.
	wikimiki.org /en/east (12235 words)

	Derived Functors (Site not responding. Last check: 2007-11-01)
		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)

	Springer Online Reference Works
		-functor (or exact functor) if it commutes with the translation functor and preserves distinguished triangles.
		be an additive left exact (respectively, right exact) functor (cf.
		be additive left exact functors between Abelian categories.
	eom.springer.de /d/d031280.htm (558 words)

	How large are left exact functors? (Site not responding. Last check: 2007-11-01)
		In contrast, for the (presheaf) category ${\cal K}=\Alg(1,1)$ of unary algebras we construct a functor from $\Alg(1,1)$ to $\Set$ which preserves finite products and is not small.
		We also describe all left exact set-valued functors as directed unions of ``reduced representables'', generalizing reduced products.
		Keywords: left exact functor, small functor, regular ultrafilter.
	www.tac.mta.ca /tac/volumes/8/n13/8-13abs.html (93 words)

	SEP: Category Theory
		Let U : Grp → Set be the forgetful functor, that is, the functor that sends to each group G its underlying set of elements U(G), and to a group homomorphism f : G → H the underlying set function U(f) : U(G) → U(H).
		Secondly, the notion of adjointness is formally equivalent to the notion of a universal morphism (or construction) and to that of representable functor.
		Thirdly, a left adjoint preserves all the colimits which exist in its domain, and, dually, a right adjoint preserves all the limits which exist in its domain.
	plato.stanford.edu /entries/category-theory (11786 words)

	[No title]
		Second, there is the problem that some of the "functors" lurking in the background are in fact only pseudo-functors - for example, the "functor" that assigns to each ring its category of finitely- generated projective modules.
		The functors H(G; M) are the derived functors* of the left exact functor M! 1If GF is finite no modifications are needed.
		In practice, the functor E will be defined on infinite extensions as well and will commute with colimits (up to weak equivalence), so we will write E F for the stalk of E.
	www.math.purdue.edu /research/atopology/Mitchell/thomason.txt (10189 words)

	[No title]
		(3.4.2) pushforward is a covariant functor (3.4.3) pushforward of a pushforward is a pushforward (3.4.4) stalk of the pushforward sheaf maps to each of the stalks in the fiber 3.5 Inverse images of presheaves.
		Because of this stalk issue, and because the stalk of a presheaf is the stalk of the associated sheaf, psi^{-1} is an exact functor from the category of sheaves on Y to the category of sheaves on X. +++ 3.8 Sheaves of pseudo-discrete spaces.
		Pushforward is left-exact because global sections are left exact of course.
	math.stanford.edu /~vakil/ega0 (5238 words)

	PlanetMath:
		left exact functor (=exact functor) owned by antizeus
		left ideal generated by (in ideal generated by a subset of a ring) owned by mathcam
		left notation (=left function notation) owned by antizeus
	planetmath.org /encyclopedia/L (2311 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)

	AdjointFunctor (Site not responding. Last check: 2007-11-01)
		is said to be a left adjoint functor to
		properties about a functor lead to formal properties of its adjoint (for example the right adjoint to a left-exact functor takes injectives to injectives).
		This pair of adjoint functors is the most commonly used and studied, and astonishingly deep facts spring from this adjoint relationship.
	202.41.85.103 /manuals/planetmath/entries/18/AdjointFunctor/AdjointFunctor.html (102 words)

	Ext functor (Site not responding. Last check: 2007-11-01)
		In mathematics, the Ext functors of homological algebra are derived functors of functors.
		Let be in and set, for fixed in.
		(This is a left exact functor (contravariant) so we want its right derived functors).
	www.gogoglo.com /wiki/en/wikipedia/e/ex/ext_functor.html (92 words)

	Math Forum Discussions
		Conclude that if B and C are submodules of a module A, >>>>>then for every module M, we have
		>>>>>functor t: r-mod -> ab such that the corresponding commutative square
		>>>>left exactness will give that the commutative square in Ab is now a
	mathforum.org /kb/thread.jspa?threadID=1124752&messageID=3691616 (555 words)

	exact - OneLook Dictionary Search
		Exact : Online Plain Text English Dictionary [home, info]
		Phrases that include exact: exact differential, long exact sequence, exact solution, fisher exact test, left exact functor, more...
		Words similar to exact: precise, accurate, claim, demand, exactable, exacted, exacter, exacting, exactness, exactor, take, call for, superaccurate, more...
	www.onelook.com /?w=exact (283 words)

	[No title]
		(nerdy2) exact functors are nice, it would be nice if given such an almost exact functor we could get an exact one out of it [15:51]
		(nerdy2) and since a short exact sequence of complexes gives a long exact sequence in homology, we have that R^i F is a delta functor [16:04]
		functor F : A -> B and A is groth.
	br.endernet.org /~loner/homcohomalg/nerdy2homalg.txt (2510 words)

	PlanetMath 2004-01-12 Snapshot: L (Site not responding. Last check: 2007-11-01)
		left and right identity laws of a monad (=monad)
		left classical ring of quotients (=classical ring of quotients)
		left identity (defined in left identity and right identity)
	simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/L.html (238 words)

	Amazon.com: A Course in Modern Algebra (Wiley Classics Library): Books: Peter Hilton,Yel-Chiang Wu (Site not responding. Last check: 2007-11-01)
		They describe a number of different algebraic domains, then emphasize the similarities and differences between them, employing the terminology of categories and functors.
		Exposition begins with set theory and group theory, and continues with coverage categories, functors, natural transformations, and duality, and closes with discussion of the two most fundamental derived functors of homological algebra, Ext and Tor.
		The notion of a group arose historically out of the attempt to extend the classical procedures for solving polynomial equations of degree 4 to equations of higher degree.
	www.amazon.com /exec/obidos/tg/detail/-/047150405X?v=glance (685 words)

	AMCA: Quantum supergroups and non-commutative analogues of vector bundles by Ruibin Zhang (Site not responding. Last check: 2007-11-01)
		We study quantum analogues of homogeneous vector bundles in the context of quantum supergoups.
		The space of holomorphic sections is promoted to a left exact covariant functor from the category of locally finite modules over a quantum parabolic subgroup to the category of locally finite modules over the quantum supergroup.
		The right derived functors of this functor provides a form of Dolbeault cohomology for quantum homogeneous supervector bundles.
	at.yorku.ca /c/a/o/h/32.htm (146 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		> 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)

	Derived categories for dummies, Part I \| The String Coffee Table
		Instead, its left derived functor is. This is usually denoted by an otimes with an L on top of it.
		Weibel is great for derived functors, but the derived category stuff is only a single chapter at the end.
		a quiver representation on a chain complex to be a functor
	golem.ph.utexas.edu /string/archives/000528.html (3457 words)

	Dror Bar-Natan: Publications
		And then a simple application of an appropriate functor (a "TQFT") to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.
		Deligne:Letter], which give the exact Kontsevich integral of the unknot and a map intertwining two natural products on a space of diagrams.
		We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of uni-trivalent diagrams.
	www.math.toronto.edu /~drorbn/LOP.html (3052 words)

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