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Topic: Preadditive categories

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	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.
		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.
		An additive category is a preadditive category with all finite biproducts.
	en.wikipedia.org /wiki/Preadditive_category (1314 words)

	Additive category - Wikipedia, the free encyclopedia
		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.
	en.wikipedia.org /wiki/Additive_category (872 words)

	Preadditive category (Site not responding. Last check: 2007-10-21)
		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.sciencedaily.com /encyclopedia/preadditive_category (1158 words)

	Preadditive category (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.worldhistory.com /wiki/P/Preadditive-category.htm (1385 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		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.
	www.informationgenius.com /encyclopedia/c/ca/category_theory.html (2864 words)

	PS Wiki Encyclopedia (Site not responding. Last check: 2007-10-21)
		In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra.
		Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field).
		That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
	70.84.119.226 /~puresear/PSWiki/index.php?title=Kernel_(category_theory) (802 words)

	Equivalence of categories - TheBestLinks.com - Axiom of choice, Algebraic geometry, Boolean algebra, Category theory, ... (Site not responding. Last check: 2007-10-21)
		One of the central themes of algebraic geometry is the duality of the category of affine schemes and the category of commutative rings.
		In pointless topology the category of spatial locales is known to be equivalent to the dual of the category of sober spaces.
		D is an equivalence of categories, and if C is a preadditive category (or additive category, or abelian category), then D may be turned into a preadditive category (or additive category, or abelian category) in such a way that F becomes an additive functor.
	www.thebestlinks.com /Equivalence_of_categories.html (1580 words)

	ring theory (Site not responding. Last check: 2007-10-21)
		A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fieldss (integral domains in which every non-zero element is invertible) act on vector spaces.
		Any ring can be seen as a preadditive category with a single object.
		It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings.
	www.yourencyclopedia.net /ring_theory.html (678 words)

	Kernel (category theory) (Site not responding. Last check: 2007-10-21)
		In category theory and its applications to other branches of mathematics, a kernel is a type of limit that generalises the notion of kernel from algebra in certain contexts.
		Kernels are familiar in many categories from abstract algebra, such as the category of groupss or the category of (left) modules over a fixed ring (including vector spaces over a fixed field).
		To be explicit, if f: A → B is a homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then K is a subalgebra of A and the inclusion homomorphism from K to A is a kernel in the categorical sense.
	www.theezine.net /k/kernel-category-theory-.html (801 words)

	Functor category - Encyclopedia Glossary Meaning Explanation Functor category (Site not responding. Last check: 2007-10-21)
		The category of presheaves of sets (abelian groups, rings) on X is then the same as the category of contravariant functors from C to Set (or Ab or Ring).
		The embedding of the category C in a functor category that was mentioned earlier uses the Yoneda lemma as its main tool.
		The category Cat of all small categories with functors as morphisms is therefore a cartesian closed category.
	www.encyclopedia-glossary.com /en/Functor-category.html (1026 words)

	Category theory (Site not responding. Last check: 2007-10-21)
		If C and D are categories, one can form the product category C × D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D.
		Using the language of category theory, many areas of mathematical study can be cast into appropriate categories, such as the categories of all sets, groups, topologies, and so on.
		These categories surely have some objects that are "special" in a certain way, such as the empty set or the product of two topologies.
	www.sciencedaily.com /encyclopedia/category_theory (3251 words)

	coequalizer (Site not responding. Last check: 2007-10-21)
		In mathematics, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category.
		In the category of sets, the coequalizer of two functions f, g : X → Y is the quotient of Y by the equivalence relation generated by the relations f(x) = g(x) for all x in X.
		In preadditive categories it makes sense to add and subtract morphisms (the hom-sets actually form abelian groups).
	www.yourencyclopedia.net /Coequalizer.html (429 words)

	Kernel (mathematics)
		There exists several notions in category theory which seek to generalize the concept of a kernel in algebra.
		The kernel pair of a morphism f is defined as a pullback of f with itself.
		Unrelated to the meanings in algebra and category theory, there is the notion of a kernel of an integral operator.
	www.mcfly.org /wik/Kernel_(mathematics) (396 words)

	math lessons - Abelian category (Site not responding. Last check: 2007-10-21)
		The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.
		If C is a small category and A is an abelian category, then the category of all functors from C to A forms an abelian category (the morphisms of this category are the natural transformations between functors).
		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).
	www.mathdaily.com /lessons/Abelian_category (917 words)

	What is Yoneda lemma? : Abaara fun facts and uncommon knowledge (Site not responding. Last check: 2007-10-21)
		A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear; examples are categories of abelian groups or modules.
		In a preadditive category, there are both a "multiplication" and an "addition" of morphisms, and that's why preadditive categories are viewed as generalizations of rings.
		The Yoneda lemma then yields the natural procedure to enlarge a preadditive category so that the enlarged version remains preadditive — in fact, the enlarged version is an abelian category, a much more powerful condition.
	www.abaara.com /pac/Yoneda_lemma (666 words)

	Pre-Abelian category - Freecyclopedia.com :: The World Bank of Knowledge (Site not responding. Last check: 2007-10-21)
		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[?].
		Every pre-Abelian category is of course an additive category, and many basic properties of these categories are described under that subject.
		In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic.
	www.freecyclopedia.com /econtents/pr/Pre-Abelian_categories.html (888 words)

	Biproduct - Encyclopedia Glossary Meaning Explanation Biproduct (Site not responding. Last check: 2007-10-21)
		In that category, the biproduct of several objects is simply their direct sum.
		But biproducts do not exist in the category of all groups; indeed, this category is not even preadditive.
		Biproducts in preadditive categories are always both products and coproducts in the ordinary category-theoretic sense; this is the origin of the term "biproduct".
	www.encyclopedia-glossary.com /en/Biproduct.html (271 words)

	Ring theory - The Encyclopedia (Site not responding. Last check: 2007-10-21)
		Maps between rings which respect the ring operations are called ring homomorphisms.
		A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.
		Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.
	www.the-encyclopedia.com /description/Ring_theory (722 words)

	Yoneda lemma Details, Meaning Yoneda lemma Article and Explanation Guide
		In mathematics, the Yoneda lemma in category theory is an abstract result on functors of the type morphisms into a fixed object.
		The Yoneda lemma suggests that instead of studying the (small) category C, one should study the category of all functors of C into Set (the category of sets with functionss as morphisms).
		The Yoneda lemma remains true for preadditive categories if we choose as our extension the category of additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of morphisms and should be thought of as forming a module category over the original category.
	www.e-paranoids.com /y/yo/yoneda_lemma.html (713 words)

	Category theory Details, Meaning Category theory Article and Explanation Guide
		General category theory — an updated universal algebra with many new features allowing for semantic flexibility and higher-order logic — came later; it is now applied throughout mathematics.
		Each category is presented in terms of its objects, its morphisms, and its composition of morphisms.
		Any directed graph generates a small category: the objects are the vertices of the graph and the morphisms are the paths in the graph.
	www.e-paranoids.com /c/ca/category_theory.html (2389 words)

	Biproduct
		In category theory and its applications to mathematics, a biproduct is a generalisation of the notion of direct sum that makes sense in any preadditive category.
		But biproducts do not exist in the category of all groups; indeed, this category isn't even preadditive.
		An additive category is a preadditive category in which every biproduct exists.
	www.fastload.org /bi/Biproduct.html (269 words)

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