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Topic: Injective objects

	Injective module - Wikipedia, the free encyclopedia
		In mathematics, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers.
		Injective modules were introduced by Reinhold Baer in 1940.
		These injective resolutions are used to define the injective dimension of a module (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as derived functors.
	en.wikipedia.org /wiki/Injective_module (986 words)

	Full and faithful functors - Wikipedia, the free encyclopedia
		In category theory, a faithful or full functor is a functor which is injective or surjective when restricted to each set of morphisms with a given source and target.
		That is, two objects X and X′ may map to the same object in D, and two morphisms f : X → Y and f′ : X′ → Y′ may map to the same morphism in D.
		However, it is neither surjective on objects or morphisms.
	en.wikipedia.org /wiki/Faithful_functor (292 words)

	Injective module -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Injective modules were introduced by (Click link for more info and facts about Reinhold Baer) Reinhold Baer in 1940.
		It is an (Click link for more info and facts about injective cogenerator) injective cogenerator in the (Click link for more info and facts about category of abelian groups) category of abelian groups, which means that it is injective and any other module is contained in a suitably large product of copies of Q/Z.
		These injective resolutions are used to define the injective dimension of a module (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as (Click link for more info and facts about derived functor) derived functors.
	www.absoluteastronomy.com /encyclopedia/i/in/injective_module.htm (1109 words)

	Injective resolution (Site not responding. Last check: 2007-10-21)
		In mathematics, an injective module is a module Q that shares certain desirable properties withthe Z-module Q of all rationalnumbers.
		It is an injective cogenerator in the category of abelian groups, which means that it isinjective and any other module is contained in a suitably large product of copies of Q/Z.
		These injective resolutions are used to define the injective dimension of amodule (the length of the shortest injective resolution ending in zeros, if such a finite resolution exists) as well as derived functors.
	www.therfcc.org /injective-resolution-113932.html (919 words)

	Injective module (Site not responding. Last check: 2007-10-21)
		In mathematics an injective module is a module Q that shares certain desirable properties with Z -module Q of all rational numbers.
		It is an injective cogenerator in the category of abelian groups which means that it is injective any other module is contained in a large product of copies of Q / Z.
		These injective resolutions are to define the injective dimension of a (the length of the shortest injective resolution in zeros if such a finite resolution as well as derived functors.
	www.freeglossary.com /Injective_resolution (1099 words)

	PlanetMath: projective object (Site not responding. Last check: 2007-10-21)
		The dual notion of a projective object is that of an injective object.
		This is version 1 of projective object, born on 2004-11-01.
		Object id is 6437, canonical name is ProjectiveObject.
	www.planetmath.org /encyclopedia/ProjectiveObject.html (53 words)

	Injective module - Information (Site not responding. Last check: 2007-10-21)
		To show that a given module is injective, the following Injective Test Lemma is useful: a left R-module Q is injective if and only if any homomorphism g : I → Q defined on a left ideal I of R can be extended to all of R.
		It is quite significant that this is also true over any ring: every module is a submodule of an injective one, or "the category of left R-modules has enough injective." To prove this, one uses the peculiar properties of the abelian group Q/Z to construct an injective cogenerator in the category of left R-modules.
		The following general definition is used: an object Q of the category C is injective if for any monomorphism f : X → Y in C and any morphism g : X → Q there exists a morphism h : Y → Q with hf = g.
	www.book-spot.co.uk /index.php/Injective_object (981 words)

	Encyclopedia: Metric space (Site not responding. Last check: 2007-10-21)
		In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects).
		In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto.
		In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values.
	www.nationmaster.com /encyclopedia/metric-space (4011 words)

	[No title]
		A rigid obstruction theory for a cofibration i is a cofibrant object W and a map a : W !
		The objects are obstruction theories for i (i.e., pairs (W, ff), where W is an object of C and ff is a function that assigns an obstruction class to each square in a functorial way).
		Here we use that the injective structure on ArC is left proper or that i and i0 are injective cofibrant so that Lemma 2.2 applies to the injective model structure on Ar C. The map p0 is not in general a fibration, so let p00be an injective fibrant replace- ment for p0.
	jdc.math.uwo.ca /papers/obstruction.txt (7257 words)

	[No title]
		The homotopy category of injectives We fix a locally noetherian Grothendieck category A. Thus A is an abelian Gro* then- dieck category and has a set A0 of noetherian objects* which generate A, that is* , every object* in A is a quotient of a coproduct of objects in A0.
		A coproduct of injective resolutions is again an injectiv* *e resolution, and the left adjoint I~ preserves all coproducts.
		The shift takes an object A to the cokernel A of a monomorphism A !* * E into an injective object E. The exact triangles are induced from short exact se* *quences in A. Proposition 7.2.
	hopf.math.purdue.edu /KrauseH/stable.txt (12046 words)

	Re: General relativity vs flat Minkowski spacetime
		This concept is formalized by saying that the >>objects of a category A are objects of a category B "equipped >>with extra structure" if there's a functor F: A -> B - in fact >>a faithful functor, but that's beside the point.
		the groupoid with just one object and one morphism is called "true" (aka "the terminal groupoid" aka "yes" aka "in") while the empty groupoid is called "false" (aka "the initial groupoid" aka "no" aka "out").
		roughly, the homotopy fiber of u over d1 is the groupoid of "objects of c equipped with designated isomorphisms from their images under u to d1"; the morphisms in the homotopy fiber are required to preserve the designated isomorphisms.
	www.lns.cornell.edu /spr/1999-12/msg0020266.html (2776 words)

	PlanetMath: Grothendieck spectral sequence
		and we can verify the hypothesis (injectives are flasque, direct images of flasque sheaves are flasque, and flasque sheaves are acyclic for the global section functor), the sequence in this case becomes:
		Cross-references: theory, algebra, Leray spectral sequence, sheaf, sequence, acyclic, injectives, global section, direct image, functor, continuous map, abelian groups, sheaves, category, topological spaces, spectral sequence, injective objects, abelian categories, between, left exact functors
		Object id is 1095, canonical name is GrothendieckSpectralSequence.
	www.planetmath.org /encyclopedia/GrothendieckSpectralSequence.html (157 words)

	Function
		By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as functions which are nowhere differentiable.
		Those functions, first thought as purely imaginary and called collectively "monsters" as late as the turn of the 20th century, were later found to be important in the modelling of physical phenomena such as Brownian motion.
		injective (one-to-one) functions send different arguments to different values; in other words, if x and y are members of the domain of f, then f(x) = f(y) if and only if x = y.
	www.fact-index.com /f/fu/function.html (2125 words)

	Algebraically compact module - InformationBlast
		A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor products C ⊗ M → C ⊗ K is injective for every right R-module C.
		More generally, every injective module is algebraically compact, for the same reason.
		Algebraically compact modules share many other properties with injective objects because of the following: there exists exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.
	www.informationblast.com /Pure-injective_module.html (574 words)

	Projective and Injective Objects (Site not responding. Last check: 2007-10-21)
		An object p is projective if, for any pair of objects a and b, and any epic morphism f from a to b, and any morphism g from p to b, there is at least one morphism h from p to a such that hf = g.
		An object is injective if each morphism upstairs implies a morphism downstairs.
		An object j is injective if, for any pair of objects a and b, and any monic morphism f from a to b, and any morphism g from a to j, there is at least one morphism h from b to j such that fh = g.
	www.mathreference.com /cat,proj.html (268 words)

	ipedia.com: Category theory Article (Site not responding. Last check: 2007-10-21)
		Then it becomes possible to relate different categories by functors, generalizations of functions which associate to every object of one category an object of another category and to every morphism in the first category a morphism in the second.
		Any monoid forms a small category with a single object x, and where every element of the monoid is a morphism from x to x (the monoid operation yields the categorical composition of morphisms).
		The solution is to characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories.
	www.ipedia.com /category_theory.html (2327 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Thus, a space D is injective in sense (1) iff for any space Y and subspace X and any continuous function f:X-->D there is a continuous extension f':Y-->D of f.
		Injective spaces are easily proved to be closed under arbitrary products and continuous retracts, which facts provide many examples once a few such spaces are known.
		Perhaps it is not so obvious, however, that injective spaces are also closed under the formation of function spaces, once the space of continuous functions is given the right topology; indeed the category of injective spaces and continuous functions is a cartesian closed category.
	www.math.psu.edu /oldColloquium/041007.html (168 words)

	[No title]
		It follows that HX is a projective object in Mod C. We shall also need to use the fact that Mod * * C is a Grothendieck category, which as far as we are concerned means that it has injec* *tive envelopes [9].
		The isomorphism class* es of indecomposable injective* objects in Mod C0 form a set since every indecompo* sable injective* C0-module arises as an injective envelope of a finitely generated C0-* *module.
		Mod C0 identifies every object in C with a C0-module which is flat and fp-i* *njective by Lemma 1.6 and Lemma 2.7.
	hopf.math.purdue.edu /KrauseH/smash.txt (13290 words)

	Injective spaces and the filter monad
		An injective space is a topological space with a strong extension property for continuous maps with values on it.
		In previous work we established an injectivity theorem for monads of this type, which characterizes the injective objects over a certain class of embeddings as the algebras.
		We thus obtain as a corollary that the injective spaces over subspace embeddings are the continuous lattices endowed with the Scott topology (Dana Scott, 1972).
	www.lfcs.informatics.ed.ac.uk /reports/98/ECS-LFCS-98-383/index.html (188 words)

	[No title]
		In A, being injective is equivalent to being a retract of such a functor.
		Moreover, the image of J is the subcategory 22 J. of injective objects in B, which is the same as the subcategory of projective ob- jects.
		Consider an object F 2 A. We know that A has enough projectives and in- jectives, so we can choose maps P f-!F g-!I where f is epic, g is monic, P is projective and I is injective.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Many categories have injective objects, but their properties depend on what families of subobjects are allowed.
		Injective spaces are also easily proved to be closed under arbitrary products and continuous retracts, which facts provide many other examples.
		Many of the properties of these spaces are provable once the spaces can be characterized as a kind of complete lattice with an appropriate, uniquely determined topology; continuity of functions then comes down to preservation of sups of directed subsets of the lattice.
	www.stanford.edu /~sommer/Scott.html (287 words)

	categories: Re: Time for functors to grow up; three queries"
		If by the psudopullback of m and G you mean the category E_1 whose objects are triples (e,gamma,d), where e and d are objects of E and D, and gamma:me-->Gd is an isomorphism in C. To see this, let E=D=C, with m and G the identity functors.
		Then an object of E_1 is just an isomorphism in C, so if C has non-identity isomorphisms, m_1 will not be injective on objects.
		If, on the other hand, by pseudopullback you mean ``anything equivalent to what I just described'', then the question of whether m_1 is injective on objects is not even well-posed, since E_1 is determined only up to equivalence.
	north.ecc.edu /alsani/ct02(1-2)/msg00065.html (652 words)

	Publications of the SPACES team
		Injectivity of real rational mappings: The case of a mixture of two gaussian laws.
		It is proved that the mapping is injective, and thus that the parameters of the mixture may be recovered from the values of the moments.
		Given a geometric object defined by rational parametric equations, we show how to compute a disjunction of implicit equations and inequations that define exactly the same object by means of regular systems.
	www-calfor.lip6.fr /~safey/Spaces/publications.html (13078 words)

	Algebraically compact module (Site not responding. Last check: 2007-10-21)
		A module homomorphism M → K is called pure injective if the induced homomorphism between the tensor products C ⊗ M → C⊗ K is injective for every right R-module C.
		More generally, every injective module isalgebraically compact, for the same reason.
		Algebraically compact modules share many other properties with injective objects because of the following: there exists exists an embedding of R-Mod into a Grothendieckcategory G under which the algebraically compact R-modules precisely correspond to the injective objects inG.
	www.therfcc.org /algebraically-compact-module-210891.html (535 words)

	A selection of what MathSciNet found when looking for: Anywhere=(tilting*)
		Following Polo, the dual Joseph modules are injective objects in subcategories of $B$-modules with suitably bounded highest weights.
		This is a refinement of an older conjecture (the injectives of $G\sb n$ have extensions to $G$-modules; this has been verified in many situations).
		For a contravariantly finite resolving subcategory the objects consist of the summands of modules which have a filtration with composition factors the minimal right approximations of the simple modules (3.8).
	www.mathematik.uni-bielefeld.de /~sek/tiltingref.html (6384 words)

	Property, Structure and Stuff
		OTOH, the Abelianization functor Groups -> Abelian groups is surjective on the objects (and on the morphisms for that matter), but groups are not Abelian groups with extra structure, because the functor isn't injective on the morphisms between a given pair.
		This means that every object of D is, not necessarily equal, but isomorphic to an object of the form U(x) for some object x of C. In general, the interesting properties of functors must be preserved by natural isomorphisms.
		Again, it's bad to care if U is injective on objects, because this property is not preserved by natural isomorphisms.
	math.ucr.edu /home/baez/qg-spring2004/discussion.html (8755 words)

	[No title]
		For the unstable modules satisfing the condition of the theorem 3 (for example, any suspension of a sub-module of H^\ast(B(Z/2)^{\oplus d}; Z/2)^{\oplus \alpha_d}, the theorem 3 gives the upper bound of the length of the gaps in the modules, which means the module does not contain arbitrary big gaps.
		In this paper we finish the proof of this conjecture, for p an odd prime, proving that there is a one-to-one correspondence between connected p-compact groups and finite reflection groups over the p-adic integers.
		Applications of such tools are briefly considered or suggested, for objects which model a directed image, or a portion of space-time, or a concurrent process.
	claude.math.wesleyan.edu /~mhovey/archive/all03 (11093 words)

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