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Topic: Abelian categories

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	Pre-Abelian category - Wikipedia, the free encyclopedia
		In mathematics, specifically in category theory, a pre-Abelian category is an additive category that has all kernels and cokernels.
		Ab is preadditive because it is a closed monoidal category, the biproduct in Ab is the finite direct sum, the kernel is inclusion of the ordinary kernel from group theory and the cokernel is the quotient map onto the ordinary cokernel from group theory.
		For example, in the category of topological Abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function.
	en.wikipedia.org /wiki/Pre-Abelian_category (882 words)

	NTU Info Centre: Abelian category (Site not responding. Last check: 2007-11-05)
		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.
	www.nowtryus.com /article:Abelian_category (893 words)

	18: Category theory, homological algebra
		Category theory, a comparatively new field of mathematics, provides a universal framework for discussing fields of algebra and geometry.
		The word "category" is used to mean something completely different in general topology.
		A full, wide-ranging text on category theory is by Borceux, Francis: "Handbook of categorical algebra", 3 vol (1: Basic category theory; 2: Categories and structures; 3: Categories of sheaves) (Encyclopedia of Mathematics and its Applications, 50-2.) Cambridge University Press, Cambridge, 1994.
	www.math.niu.edu /~rusin/known-math/index/18-XX.html (286 words)

	Semi-Abelian Categories - Janelidze, Marki, Tholen (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object.
		2: protomodular and semi-abelian categories (context) - Borceux, Bourn - 2002
		5 nilpotency and solvability in categories (context) - Huq - 1968
	citeseer.ist.psu.edu /janelidze00semiabelian.html (995 words)

	[No title]
		Toposes and abelian categories are striking for the number of elementary properties they have in common (monic+epi = iso, mono/epi is a unique factorization system, etc. etc.) and the paucity of their common models, namely just the final category.
		To the extent that both toposes and abelian categories share much pleasant structure, the models of the intersection of their theories, for a suitable choice of language, would seem to be a nice class in its own right.
		For abelian cats the lemma remains true when one relaxes the hypothesis from "a pair of maps one of which is monic" to "a pair of maps that are jointly monic." Such a lemma is very wrong for topoi.
	www.mta.ca /~cat-dist/catlist/1999/atcat (5103 words)

	[No title]
		An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived cat- egory of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra.
		A cofibrantly generated model category is a model category M for which there exist sets I and J of morphisms with domains that are small relative to I-cof and J -cof, respectively, such that I-cof is the category of cofibrations and J -cof is the category of trivial cofibrations.
		We are concerned with two projective classes on the category A. The first is the categori- cal projective class C whose projectives are summands of free modules, whose exact sequences are the usual exact sequences, and whose epimorphisms are the surjections.
	jdc.math.uwo.ca /papers/relative.txt (10317 words)

	notes on: Categories for the Working Mathematician
		First he points out that Category Theory is "to discuss properties of totalities" such as the "set" of all groups.
		These means that category theory would like to be able to use an unrestricted principle of comprehension, were it not that this is known to give rise to problems of consistency (e.g.
		He then draws the distinction between small and large categories, the former being the ones whose sets of morphisms and of objects are members of U (or in NBG are sets rather than classes).
	www.rbjones.com /rbjpub/philos/bibliog/macla71.htm (303 words)

	categories: Re: Abelian Topological Groups (Site not responding. Last check: 2007-11-05)
		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]
		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.
		Anything as elegant as the Abelian category axioms--though of course elegance is often in the eye of the beholder.
		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)

	categories: Abelian Topological Groups (Site not responding. Last check: 2007-11-05)
		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.
		Of course, having a topology as an object in the abelian category means we have to have objects in the category other than abelian groups.
		I seem to be using the category of complete, hausdorff uniform spaces as a base category.
	north.ecc.edu /alsani/ct01(1-4)/msg00099.html (189 words)

	Lucian Ionescu's Stuff
		In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups.
		In derived categories the cone of a map is a canonical generator for the corresponding cokernel ideal.
		The ``space-time'' should be reconstructed as a prime spectrum of a category, as part of a ``correspondence principle'', and should not be considered a primary concept.
	www.ilstu.edu /~lmiones/research.htm (1098 words)

	Preadditive category : Additive functor (Site not responding. Last check: 2007-11-05)
		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)

	UCL/AGEL - (Site not responding. Last check: 2007-11-05)
		Strongly connected with internal categories, symmetric categorical groups have been studied as a 2-dimensional analogous of abelian groups, obtaining applications to ring theory, group extensions, homotopy groupoids and factorization systems.
		In particular, the category of Q-sets (Q being a quantale) and the étale morphisms between quantales have been studied, and representation theorems for C*-algebras and rings via their quantales of ideals have been obtained.
		We have exhibited a characterization of those categories in which the groups of automorphisms are representable (like the category of all groups).
	www.math.ucl.ac.be /AGEL/AGELrech1.html (1085 words)

	Practical Foundations of Mathematics
		Conversely, any category with a zero object and biproducts is CMon-enriched, ie the hom-sets carry a commutative monoid structure for which composition is linear in each argument separately (Exercise 5.20).
		5.4.5 Homological algebra was the progenitor of category theory.
		Abelian categories are covered thoroughly in [Fre64], [ML71], [FS90] and in any modern homology text.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s54.html (1889 words)

	Category Theory
		Category theory reveals that many of these constructions are in fact special cases of objects in a category with what is called a "universal property".
		For it is in his thesis that Lawvere proposed the idea of developing the category of categories as a foundation for category theory, set theory and, thus, the whole of mathematics, as well as using categories for the study of theories, that is the logical aspects of mathematics.
		Given these simple facts, it remains to be seen whether category theory should be "on the same plane", so to speak, with set theory, whether it should be considered seriously as providing a foundational alternative to set theory or whether it is foundational in a different sense altogether.
	plato.stanford.edu /entries/category-theory (7029 words)

	[No title]
		The original motivating example of a model category is the catego* ry Top of topological spaces, with W the class of homotopy equivalences, C the c *lass of cofibrations, and F the class of (Hurewicz) fibrations (cf.
		If M is an abelian category with enough injectives, there is a model category structure on cM with WcM the class of cohomology isomorphisms, CcM the maps which are one-to-one in positive degrees, and FcM the surject* *ive maps with injective kernel.
		It should perhaps be observed that the situation for an abelian ca* tegory M, in which both left and right derived functors may be defined, is anomolous: it arises because M may be viewed either as a category* of universal algebras or as* * a category of universal coalgebras, over itself.
	hopf.math.purdue.edu /Blanc/Blanc_model.txt (7029 words)

	M. SC. THESIS
		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.
		Except for 7.10 (ii), which was inspired by the similar result for abelian categories, all the results in section 7 are from [Cassidy, Hébert and Kelly, Reflective subcategories, localizations and factorization systems, J. Austral.
	www.mat.uc.pt /~picado/publicat/Summary.html (1547 words)

	On the Freyd Categories of an Additive Category (Site not responding. Last check: 2007-11-05)
		On the Freyd Categories of an Additive Category
		$\mathcal B(\C)$, is the reflection of $\C$ in the category of additive categories with cokernels, resp.
		The purpose of the paper is to study further the Freyd categories and to indicate their applications to the module theory of an abelian or triangulated category.
	emis.u-strasbg.fr /journals/HHA/volumes/2000/n11/abstract.htm (134 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)

	Negative K-theory of Derived Categories, by Marco Schlichting (Site not responding. Last check: 2007-11-05)
		We define negative K-groups for exact categories and for ``derived categories'' in the framework of Frobenius pairs, generalizing definitions of Bass, Karoubi, Carter, Pedersen and Thomason.
		We show that the first negative K-group of an abelian category vanishes, and that, in general, negative K-groups of a noetherian abelian category vanish.
		In the appendix, we replace Waldhausen's cylinder functor by a slightly weaker form of non-functorial factorization which is still sufficient to prove his approximation and fibration theorems.
	www.math.uiuc.edu /K-theory/0636 (156 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The machinery needed to define a derived category in full generality tends to obscure the simplicity of the phenomena.
		The class of abelian categories is not closed under many important constructions.
		Thus the category of projective objects or the category of filtered objects of an abelian category are no longer abelian in general.
	www.elsevier.com /homepage/saj/523281/h19.htm (346 words)

	Oct 13-17 VU Math Events
		We will explore some of these: comparing how category theory distinguishes between regular and normal epimorphisms, and algebra distinguishes between congruences and ideals; how internal precrossed modules and internal reflexive graphs compare to semi-congruences and ideals; how the algebraic notion of "clot" is translated categorically [Janelidze-Marki-Ursini (2003)].
		By elaborating on their results I will show that there is a full and faithful functor from the category of locally finite trees and classes of quasi-isometries to the category of compact ultrametric spaces and quasi-conformal Holder homeomorphisms.
		The multigrid method is a powerful tool to solve algebraic systems of equations arising in many applications and it is known to be among a few methods to provide an optimal complexity in terms of arithmetic operations per unknown.
	math.vanderbilt.edu /~calendar/archive/2003/10_13.html (928 words)

	[No title]
		Acknowledgments: The Morita theory in stable model categories which I descri* be in Section 4 is based on joint work with Brooke Shipley spread over many years and * several papers; I would like to take this opportunity to thank her for the pleasant and* * fruitful collaboration.
		Since the category of right Rop-modules is isomorphic to the cate* gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a nd then they provide the equivalence of categories* between Mod-Ropand Mod-Sop.
		The bimodules which induce the equivalences of module categories ca* n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec *tors' (or n x 1 matrices) respectively.
	hopf.math.purdue.edu /Schwede/Morita.txt (4235 words)

	Open Directory - Science: Math: Algebra: Category Theory (Site not responding. Last check: 2007-11-05)
		Category Theory - This expository article is an entry in the Stanford Encyclopedia of Philosophy.
		The Computational Category Theory Project - The aim of the project is the development of software on a wide variety of platforms for computing with mathematical categories and associated algebraic structures.
		CT Category Theory - Section of the e-print arXiv dealing with category theory, including such topics as: enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
	dmoz.org /Science/Math/Algebra/Category_Theory (327 words)

	Abstracts
		Associated with any class of objects of an Abelian category is the class of proper (with respect to that class), short exact sequences.
		In the category of Abelian p-groups, a complete description of all proper generators remains unknown, but it is shown that the class includes all non-reduced generators, all generators
		Similar theorems are obtained for a number of other classes of cotorsion groups, including the class of all reduced p-adic cotorsion groups and the class of all reduced torsion free p-adic cotorsion groups.
	www.math.nmsu.edu /~hardy/carolabstracts.html (2579 words)

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