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Topic: Right exact functor

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	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
		Sheaf cohomology arises as the right derived functors of the global section functor on sheaves.
		Étale cohomology arises as the right derived functors of the global sections functor on the category of étale sheaves; this example includes as special cases the previous two.
		This is version 17 of derived functor, born on 2003-02-10, modified 2006-05-15.
	planetmath.org /encyclopedia/DerivedFunctor.html (382 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
		This is a left-exact functor, and its right derived functors are the group cohomology functors, typically written as H
	en.wikipedia.org /wiki/Derived_functor (1209 words)

	[No title]
		Exactness of f0implies that f is cohom* *ological, since h is cohomological.
		In fact, both functors are right exa* ct, pre- serve coproducts, and coincide on the full subcategory of finitely generated pr *ojective objects in Mod C0.
		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)

	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)

	Functor Object
		LazyEvaluation makes it impossible to know the exact order in which operations will be executed and in fact makes it possible for some operations not to be executed at all.
		In CategoryTheory, the field of mathematics where the term Functor originates, a Functor is a mapping from one Category to another that respects Morphisms [functions].
		Functors may also inform a factory about what they can do - either by proxy, or directly, allowing the factory to select the "optimum" functor from a collection of functors, using hints provided by the invoker of the factory...
	c2.com /cgi/wiki?FunctorObject (1812 words)

	Derived Functors
		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)

	Homological Algebra 2
		By definition, the horsehoe resolution $P_B$ is a free resolution of $B$, together with a short exact sequence of chain complexes $$0 \longrightarrow P_A \longrightarrow P_B \longrightarrow P_C \longrightarrow 0$$ extending the original short exact sequence.
		The horseshoe resolution is used to construct the connecting homomorphism for any left-exact, or contravariant right exact functor, e.g.
		We wish to determine from the exact sequence $$ 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0,$$ the matrix $C_1 \rightarrow A_0$, where $A_0$ is the free module mapping onto $A$, and $C_1 \rightarrow C_0$ is the presentation matrix of $C$.
	www.mathematik.uni-ulm.de /help/Macaulay2-0.8.42/1202.html (1049 words)

	[No title] (Site not responding. Last check: 2007-10-11)
		Every functor from a suitable subcategory of A to B has a unique extension to a right exact functor from A to B. This leads to the definition of tensor induction for Mackey functors, associated to any finite biset.
		This is well behaved with respect to the usual constructions on Mackey and Green functors (direct sums, tensor products).
		This gives also a generalized tensor induction for p-permutation modules and algebras.
	www.maths.abdn.ac.uk /~bensondj/papers/b/bouc/tens_ind.data (97 words)

	Inductive Limits (Site not responding. Last check: 2007-10-11)
		Such an inductive system is said to have inductive (or direct) limit, denoted by
		Of course such an object need not exist, but if for every inductive system, we can form inductive limits, then this is a well defined (covariant) functor
		The next theorem states condition under which we can hope for existence of direct limits and exactness properties of this functor (assuming of course that underlying category is abelian).
	www.imsc.res.in /~sgautam/main/node4.html (92 words)

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