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Topic: Category of abelian groups

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	Category of abelian groups - Wikipedia, the free encyclopedia
		This is the prototype of an abelian category.
		The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
		In Ab, the notion of kernel in the category theory sense coincides with kernel in the algebraic sense, i.e.: the kernel of the morphism f : A → B is the subgroup K of A defined by K = {x in A : f(x) = 0}, together with the inclusion homomorphism i : K → A.
	en.wikipedia.org /wiki/Category_of_abelian_groups (565 words)

	Abelian category - Wikipedia, the free encyclopedia
		The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
		Important theorems that apply in all abelian categories include the five lemma (and the short five lemma as a special case), as well as the snake lemma (and the nine lemma as a special case).
		Abelian categories were introduced by Alexander Grothendieck in the middle of the 1950s in order to unify various cohomology theories.
	en.wikipedia.org /wiki/Abelian_category (922 words)

	Abelian group - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-20)
		In mathematics, an abelian group, also called a commutative group, is a group (G, ) such that a b = b * a for all a and b in G.
		This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
		The abelian group, together with group homomorphisms, form a category, the prototype of an abelian category.
	www.newlenox.us /project/wikipedia/index.php/Abelian_group (824 words)

	PlanetMath: category theory (Site not responding. Last check: 2007-10-20)
		Category theory gives us tools for analyzing such functors: we can talk about natural transformations of functors, and in fact we can use these to assemble the category of functors from one category to another into a category, provided certain set-theoretic constraints are met (universes are a tool used to address these set-theoretic difficulties).
		The fundamental theorem of Galois theory is that the functor from a subgroup of the Galois group of a field to its fixed field is an equivalence of categories.
		Central to the field is the notion of an abelian category.
	planetmath.org /encyclopedia/CategoryTheory.html (1642 words)

	PlanetMath: category of pointed topological spaces (Site not responding. Last check: 2007-10-20)
		This yields a functor from the category of pointed topological spaces to the category of groups.
		Cross-references: category of topological spaces, loop space, abelian groups, map, higher homotopy groups, groups, functor, group homomorphism, fundamental group, zero object, singleton, homeomorphism, isomorphic, category, continuous map, morphism, topological space
		This is version 3 of category of pointed topological spaces, born on 2003-10-15, modified 2004-01-24.
	planetmath.org /encyclopedia/CategoryOfPointedTopologicalSpaces.html (196 words)

	Adjoint functors (Site not responding. Last check: 2007-10-20)
		Similarly, the group ring construction yields a functor from groupss to rings, left adjoint to the functor that assigns to a given ring its group of units.
		In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum.
		The article on Stone duality describes an adjunction between the category of topological spaces and the category of sober spaces that is known as soberification.
	www.bidprobe.com /en/wikipedia/a/ad/adjoint_functors.html (3145 words)

	Abelian category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		A category is (Click link for more info and facts about preadditive) preadditive if it is (Click link for more info and facts about enriched) enriched over the (Click link for more info and facts about monoidal category) monoidal category Ab of (A group that satisfies the commutative law) abelian groups.
		The category of all (Click link for more info and facts about finitely generated abelian group) finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
		Abelian categories were introduced by (Click link for more info and facts about Alexander Grothendieck) Alexander Grothendieck in the middle of the 1950s in order to unify various (Click link for more info and facts about cohomology) cohomology theories.
	www.absoluteastronomy.com /encyclopedia/a/ab/abelian_category.htm (1378 words)

	Abelian category
		In mathematics, an abelian category is a certain kind of category in which morphisms and objects can be added and in which kernelss and cokernels exist and have the usual properties.
		The motivating prototype example of an abelian category is the category of abelian groups.
		This means that all morphism sets are abelian groups and the composition of morphisms is bilinear.
	www.brainyencyclopedia.com /encyclopedia/a/ab/abelian_category.html (995 words)

	Category theory - FreeEncyclopedia (Site not responding. Last check: 2007-10-20)
		Category theory is also used in a foundational way in functional programming, for example to discuss the idea of typed lambda calculus in terms of cartesian-closed categories.
		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.
		One of the central themes of algebraic geometry is the equivalence of the category C of affine schemes and the category D of commutative rings.
	openproxy.ath.cx /ca/Category_theory.html (2075 words)

	Exact sequence : QuicklyFind Info (Site not responding. Last check: 2007-10-20)
		In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next.
		To be precise, fix an Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field) or some other category with kernels and cokernels (such as the category of all groups).
		where 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z/2Z is given by reducing integers modulo 2.
	www.quicklyfind.com /info/Exact_sequence.htm (1011 words)

	categories: Re: Abelian Topological Groups (Site not responding. Last check: 2007-10-20)
		Moreover, although a weaker topology (or an abelian group with a weaker topology, which is what I assume is meant) is certainly a subobject, it is not regular, which every subobject in an abelian category must be.
		In fact, the only abelian categories of topological abelian groups I am aware of are the discrete groups and the dual category of compact groups.
		The idea is that the > quotients of such a group, in the abelian category, would be completions > of the group with respect to topologies coarser than the given one.
	north.ecc.edu /alsani/ct01(5-8)/msg00001.html (352 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Subject: categories: Chu(Ab,circle) is abelian The category of topological abelian groups is not abelian.
		The conditions are self dual and a limit is computed in a Chu category as the limit of its first component and colimit of its second and all the required isomoprhisms remain isomorphisms.
		The contrast with the case of topological and that of localic abelian groups is striking.
	www.mta.ca /~cat-dist/catlist/1999/chu-abelian (664 words)

	[No title]
		At least: the category of all groups, the category of groups with a fixed set of operators on them and homomorphisms prserving the operators, and the same for Abelian groups in place of all groups.
		You could axiomatize the category of all groups by, in effect, axioms for the category of sets (to be construed as free groups) plus the quotients given by the triple for groups over sets.
		But the key seems to be that the category of groupoids is cartesian closed and its insertion into the category of categories preserves exponentials--the prominent fact that a natural transformation with all components iso is a natural iso.
	www.mta.ca /~cat-dist/catlist/1999/nonabel (1055 words)

	Preadditive category : Additive functor (Site not responding. Last check: 2007-10-20)
		A preadditive category is a category that is enriched over the monoidal category[?] of abelian groups.
		An additive category is a preadditive category with all finite biproducts.
		A pre-Abelian category is an additive category with all kernels and cokernels.
	www.city-search.org /ad/additive-functor.html (1378 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Generalizations of purity in the category of abelian groups and in module categories have many applications and are really tools of homological algebra.
		This construction was made in [] to give a unified approach to homological algebra in pre-abelian categories which allows to include the approaches in [] in the framework of a single theory.
		Section is devoted to relative homological algebra in module categories and we discuss recent results on the classification of inductively closed proper classes which are closely related with algebraically compact modules.
	www.elsevier.com /homepage/saj/523281/h17.htm (527 words)

	Articles - Abelian group (Site not responding. Last check: 2007-10-20)
		The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules.
		When studying abelian groups in their own right, the additive notation is usually used.
		Every ring is an abelian group with respect to its addition operator.
	www.lastring.com /articles/Non-abelian?mySession=c2f43531e6199808d31fa4aeacef9c19 (846 words)

	[No title]
		Part of the interest of these results is that the family of categories equi* valent to that of crossed complexes can be regarded as a foundation for a non-abelian approach to algebra ic topology and the cohomology of groups*.
		Internally to the category of groups, these are more complicated; but internally to the categ* ory of abelian* groups they are again equivalent to morphisms of abelian groups.
		B3 ' B4 : Let K be a cubical abelian group with connections, in the sense o* *f [4].
	hopf.math.purdue.edu /BrownR-Higgins/cubabgp3.txt (946 words)

	Introduction (Site not responding. Last check: 2007-10-20)
		In the case of the category of permutation groups and the category of soluble groups defined by a power-conjugate presentation, all groups in the category are finite.
		However, the finitely-presented group category, the polycyclic group category, the abelian group category and the matrix group category contain both finite and infinite groups.
		In the case of the abelian group category and the matrix group category, a large number of functions are available for finite groups only.
	www.math.niu.edu /help/math/magmahelp/text235.html (208 words)

	[No title]
		1 2 J. category A of additive functors from F to the category Ab of Abelian groups, with emphasis on the homology theories.
		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.
		The subcategory consisting of towers of finite spectra is equivalent to the opposite of the category of homology theories with countable coefficients.
	jdc.math.uwo.ca /papers/phantoms.txt (11288 words)

	AMCA: Duality for Convergence Abelian Groups by M. Montserrat Bruguera
		Examples of reflexive groups which are not locally compact are known from the late forties.
		If G is a LCA group, the continuous convergence structure in \GammaG is precisely the convergence given by the compact open topology [3], thus, the ''convergence dual'' and the ordinary dual are identical.
		G is an isomorphism in the category CONABGRP.
	at.yorku.ca /c/a/a/h/10.htm (846 words)

	Abelian categories
		It might refer to the fact that the homsets in an abelian category are abelian groups.
		It might refer to the fact that the category of abelian groups is a very nice example of an abelian category.
		You seem to be eager to learn what an abelian category is. That's a noble ambition, so why don't we concentrate on that for a while instead of all the fancier stuff.
	www.lns.cornell.edu /spr/1999-12/msg0020321.html (1010 words)

	[No title]
		to abelian groups; we often consider presheaves with values in different categories, such as the category of sets, or the category of commutative rings.
		is a (normal) subgroup of the (abelian) group of all locally constant functions.
		The restriction maps are induced from the usual restrictions of locally constant functions, and are well-defined and functorial because of the universal properties of quotient group homomorphisms.
	odin.mdacc.tmc.edu /~krc/agathos/sheaf.html (579 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		What's known is that "all Abelian groups" is much too murky a family of objects to permit this kind of structural theorem.
		These completely characterize the group up to isomorphism _if_ the group is countable and contains no copy of what you call A_p/Z (the direct limit of the cyclic groups Z/(p^k Z).) I don't really think there is a good characterization of torsion-free abelian groups.
		It is similarly true that one can make some headway characterizing Abelian groups with some other limiting condition, such as divisible groups, groups of bounded height, matrix groups, etc. Of course there's more to do in a field than simply find a structure theorem for the objects.
	www.math.niu.edu /~rusin/known-math/99/ab_gps (613 words)

	[No title]
		This was done in such a way that for a group G one could recover the topos of G-sets (and hence the group itself) from the category of linear representations.
		A category C is definable in a closed category D when it embeds fully (as a category) in every self-dual closed category embedding D (as a closed category).
		A category C is definable in a closed category V when it embeds fully (as a category) in every self-dual closed category embedding V (as a closed category).
	www.mta.ca /~cat-dist/catlist/1999/set-abel-group (2757 words)

	M. SC. THESIS
		The concept of torsion had its origin in the theory of abelian groups.
		As Herrlich and Strecker refer in [Category Theory, Allyn and Bacon, Boston, 1973], quoting Bass, "Virtually all algebraic notions in Category Theory are parodies of their parents in the most classical of categories...
		Japan 17 (1965) 30-35], we present, with some detail, a classification in the category of abelian groups of all torsion subcategories contained in the subcategory of torsion groups (in the classical sense) and of all hereditary torsion subcategories.
	www.mat.uc.pt /~picado/publicat/Summary.html (1547 words)

	A Useful Category For Mixed Abelian Groups. (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		All the useful categories in the study of the mixed abelian groups (e.g.
		We introduce a new category denoted A which ignores the torsion-freeness and could characterize some classes of nonsplitting mixed groups with the aid of Walk.
		Introduction The categories Warf, first introduced as H in [7] and Walk, first introduced as C in [2] have useful applications in the theory of the mixed abelian groups.
	citeseer.ist.psu.edu /350920.html (281 words)

	Office of the Provost and Chief Academic Officer
		Groups, subgroups, cyclic groups, quotient groups, Lagranges Theorem, permutation groups, homomorphism and isomorphism theorems, Cayley's theorem, rings, subrings, ideals, fields, homomorphism and isomorphism theorems.
		Sylow's theorems for finite groups, p-groups, abelian groups, group action on sets, domains, prime and maximal ideals, unique factorization domain.
		Groups, group actions on sets, structure of finitely generated abelian groups, category theory, exact sequences, rings, P.I.D’s, modules, direct sum and direct product, Hom and duality, tensor products, projective, injective, flat and free modules.
	www.provost.howard.edu /provost/bulletin2/g/v2gmath_a.htm (356 words)

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