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Topic: Pre Abelian categories

	Preadditive category - Wikipedia, the free encyclopedia
		Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object.
		An additive category is a preadditive category with all finite biproducts.
		A pre-Abelian category is an additive category with all kernels and cokernels.
	en.wikipedia.org /wiki/Preadditive_category (1314 words)

	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)

	Cokernel (Site not responding. Last check: 2007-10-21)
		In the category of groups, the cokernel of a group homomorphism f : G → H is the quotient of H by the normal closure of the image of f.
		In such a category, the coequalizer of two morphisms f and g (if it exists) is just the cokernel of their difference:\n:coeq(f, g) = coker(g - f) In a pre-abelian category (a special kind of preadditive category) the existence of kernels and cokernels is guaranteed.
		In such categories the image and coimage of a morphism f are given by\n:im(f) = ker(coker f)\n:coim(f) = coker(ker f) Abelian categories are even better behaved with respect to cokernels.
	encyclopedia.codeboy.net /wikipedia/c/co/cokernel.html (456 words)

	Preadditive category -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A preadditive category is a (A general concept that marks divisions or coordinations in a conceptual scheme) category that 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 of (A group that satisfies the commutative law) abelian groups.
		In contrast, the category of all ((chemistry) two or more atoms bound together as a single unit and forming part of a molecule) groups is not closed.) See (Click link for more info and facts about medial category) medial category.
		If C and D are categories and D is preadditive, then the (Click link for more info and facts about functor category) functor category Fun(C,D) is also preadditive, because (Click link for more info and facts about natural transformation) natural transformations can be added in a natural way.
	www.absoluteastronomy.com /encyclopedia/p/pr/preadditive_category.htm (1551 words)

	Preadditive category
		A preadditive category is a category that is enriched over the monoidal category of abelian groups.
		and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object.
		When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of kernel of a homomorphism, if one identifies the ordinary kernel
	www.brainyencyclopedia.com /encyclopedia/p/pr/preadditive_category.html (1147 words)

	Category theory (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.
		Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders MacLane in 1945.
		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.
	www.sciencedaily.com /encyclopedia/category_theory (3261 words)

	Additive category - Wikpedia (Site not responding. Last check: 2007-10-21)
		Ab is preadditive because it is a closed monoidal category, and the biproduct in Ab is the finite direct sum.
		Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subject.
		In fact, it is a theorem that all adjoint functors between additive categories must be additive functors, and most interesting functors studied in all of category theory are adjoints.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Additive_category (900 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)

	Preadditive category - Wikpedia (Site not responding. Last check: 2007-10-21)
		If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab.
		That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism.
		When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.
	www.bostoncoop.net /~tpryor/wiki/index.php?title=Preadditive_category (1295 words)

	[No title]
		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)

	categories: Mac Lane and Abelian Categories (Site not responding. Last check: 2007-10-21)
		He says an "Abelian category" is a category with a generator and a cogenerator and with zero object and biproducts (which he calls "free-and-direct products).
		He proves that in such a category, the arrows from any object A to another B form an additive commutative monoid (he actually says "semigroup", p.511); and that any such category is isomorphic to a category of commutative monoids (p.512).
		A category with submaps is a category with a distinguished class of monics satisfying certain axioms (which imply that every split monic is in the class, p.499).
	north.ecc.edu /alsani/ct99-00(8-12)/msg00209.html (632 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)

	Pre-Abelian category (Site not responding. Last check: 2007-10-21)
		A pre-Abelian category is an additive category that has all kernels and cokernelss.
		Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functionss.
		: C → D between preadditive categories that acts as a group homomorphism on each hom-set.
	www.sciencedaily.com /encyclopedia/pre_abelian_category (874 words)

	Pre-Abelian category (Site not responding. Last check: 2007-10-21)
		Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an Abelian group by item 1; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.
		We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
		First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism on each hom-set.
	www.worldhistory.com /wiki/P/Pre-Abelian-category.htm (956 words)

	Mat 1191S - Introduction to sheaves and schemes. (Site not responding. Last check: 2007-10-21)
		The second part of the course will be dedicated to scheme theory.
		Topics covered in Mat 1191S are: category theory, sheaves, schemes, applications.
		categories and functors; morphisms of functors; representable functors
	www.math.toronto.edu /~kc/1191.html (135 words)

	[No title]
		We also show that the homotopy categories associated to the two categories are equivalent to the homotopy categories of simplicial presheaves and homotopy 2-types, respectively.
		The idea behind this thesis is to stabilize the category of crossed complexes, as it is an interesting approximation to the category of simplicial sets, reflecting certain, though not all, nonabelian homotopical information concerning simplicial sets.
		The precise meaning of `small generator' depends on the context, be it an abelian category, a derived category or a stable model category.
	www.lehigh.edu /~dmd1/ho1024.txt (1659 words)

	[No title]
		Comodule categories and the geometry of the stack of formal groups N. Naumann Abstract We generalise recent results of M. Hovey and N. Strickland on comodule cate* *gories for Landweber exact algebras using the formalism of algebraic stacks.
		It is "well-known" that the category of comodules is equivalent to the * category of quasi- coherent sheaves of modules on an algebraic stack associated to the flat Hopf a *lgebroid.
		This implies that their catego* ries of representa- tions over C are equivalent as abelian* categories.
	hopf.math.purdue.edu /Naumann/comodlandweber.txt (5917 words)

	[No title]
		A large number of examples of kinds of structured category is presented showing that the structured categories are selected from among the corresponding generalized sketches as the models of a set of sketch-entailments.
		Along the same lines, the categories of processes modulo the observational and the branching congruences, with the suitable simulations as morphisms again, are shown to be isomorphic with certain subcategories of the category of irredundant trees.
		In particular, we present a category where the path order is not omega-complete, but in which the constructions of domain theory (as, for example, the existence of uniform fixed-point operators and the solution of domain equations) are possible.
	www.mta.ca /~cat-dist/catlist/preprints/paper-availability (17382 words)

	Lucian M. Ionescu (Site not responding. Last check: 2007-10-21)
		The collaboration focused on the problem of spectra in additive categories, which aims to implement a mathematical model for the physical idea that paths are the “points” of a space (spectrum) associated to the “category of Feynman diagrams”.
		F’99: I begun studying the spectra of additive categories in terms of ideals and the relation with the existing definitions, advised by A. Rosenberg.
		The research on the correspondance between set theoretical notions and category theory concepts, motivated by the correspondence between the cohomology of monoidal categories and nonabelian group cohomology, was finalized as [1,13].
	www.ilstu.edu /~lmiones/ProdRep04CV.htm (3585 words)

	Recent preprints - Maria Manuel Clementino (Site not responding. Last check: 2007-10-21)
		Abstract: We introduce a relativized notion of (semi)normalcy for categories that come equipped with a proper stable factorization system, and we use radicals and normal closure operators in order to study torsion theories in such categories.
		We pay particular attention to the homological and, for our purposes more importantly, normal categories of topological algebra, such as the category of topological groups.
		But our applications go far beyond the realm of these types of categories, as they include, for example, the normal, but non-homological category of pointed topological spaces, which is in fact a rich supplier for radicals of topological groups.
	www.mat.uc.pt /~mmc/preprints (146 words)

	Bmw Intake Manifold (Site not responding. Last check: 2007-10-21)
		Bernhard Riemann was the first to do extensive work that really required a generalization of Manivold s to higher dimensions.
		Abelian varieties were at that time already implicitly known, as complex Mqnifold s.
		Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, were also naturally Mahifold theories, with aconcept of generalized coordinates.
	www.super8filmmaking.com /tail/24350-bmw-intake-manifold.html (633 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Just a thought about Vaughan's original question: the class of toposes (and that of pretoposes) is stable under slicing, as are all the `exactness properties' that they share with abelian categories.
		Slices of abelian categories aren't abelian; but, thanks to Aurelio Carboni, we know how to characterize them.
		So we ought surely to be looking for a common generalization, not of toposes and abelian categories, but of (pre)toposes and affine categories in Aurelio's sense.
	www.mta.ca /~cat-dist/catlist/1999/prattsli (199 words)

	Preadditive category : Additive functor (Site not responding. Last check: 2007-10-21)
		Current city Street: Preadditive category : Additive functor <
		In contrast, the category of all groups isn't closed.)
		When specializing to the preadditive categories of abelian groups or modules over a ring, thisn'tion of kernel coincides with the ordinary notion of kernel of a homomorphism, if one identifies the ordinary kernel
	www.city-search.org /ad/additive-functor.html (1378 words)

	Many Familiar Categories Can Be Interpreted as Categories of Generalized Metric Spaces - Heitzig (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Jobst Heitzig, Many familiar categories can be interpreted as categories of generalized metric spaces, to appear in Appl.
		@misc{ heitzig-many, author = "J. Heitzig", title = "Many familiar categories can be interpreted as categories of generalized metric spaces", text = "Jobst Heitzig, Many familiar categories can be interpreted as categories of generalized metric spaces, to appear in Appl.
		76 Abstract and concrete categories (context) - amek, Herrlich et al.
	citeseer.ist.psu.edu /heitzig99many.html (520 words)

	abelian categories - OneLook Dictionary Search (Site not responding. Last check: 2007-10-21)
		We found one dictionary with English definitions that includes the word abelian categories:
		Tip: Click on the first link on a line below to go directly to a page where "abelian categories" is defined.
		Phrases that include abelian categories: pre abelian categories
	www.onelook.com /?w=abelian+categories (79 words)

	The Determinant Line Bundle over Moduli Spaces of Instantons on Abelian Surfaces - Maciocia (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		We study the determinant line bundle over moduli space of stable bundles on abelian surfaces.
		We extend Mukai's version of the Parseval Theorem to L 2 metrics on cohomology groups.
		Maciocia, The determinant line bundle over moduli spaces of instantons on abelian surfaces, Math.
	citeseer.ist.psu.edu /maciocia94determinant.html (528 words)

	Stefan Muller-Stach (Site not responding. Last check: 2007-10-21)
		algebra over the category of structured spectra), we are developing a kind of algebraic geometry in homotopical contexts (like for example in the category C(k) of complexes of k-modules).
		When the model category has a compatible monoidal structure we also define, following Simpson, a notion of geometric or algebraic stack on it.
		This could also be seen as a first step in a more general theory of algebraic geometry over monoidal infinity categories; as an instance of this we compare and unify the previously described approaches using Simpson's Segal categories.
	www.informatik.uni-mainz.de /~stefan/homotopy.html (616 words)

	Program
		Wednesday, July 25, is arrival, orientation, and registration day but there will be a reception in the evening, which is free for all registered participants.
		Algebraic features of the abelian groups (e.g., free rank, p-ranks, etc.) preserved under homeomorphisms w.r.t.
		On determining the classification difficulty of countable torsion free abelian groups
	www.math.hawaii.edu /~adolf/Meeting/wwwprogram.htm (258 words)

	Books, Bytes and Beyond (Site not responding. Last check: 2007-10-21)
		You can use the menu on the left to select different subject categories.
		You can search each major category by subject.
		Click on a subject and sub-heading and the books on that subject will display.
	booksbytesandbeyond.com /Merchant2/merchant.mvc?store_code=1&... (123 words)

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