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Topic: Cokernel category theory

	Normal morphism - Wikipedia, the free encyclopedia
		In category theory and its applications to mathematics, a normal monomorphism or normal epimorphism is a particularly well-behaved type of morphism.
		A normal category is a category in which morphisms are normal, whenever reasonable.
		The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.
	en.wikipedia.org /wiki/Binormal (292 words)

	Cokernel - Wikipedia, the free encyclopedia
		In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f.
		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 a pre-abelian category (a special kind of preadditive category) the existence of kernels and cokernels is guaranteed.
	www.wikipedia.org /wiki/Cokernel (425 words)

	Learn more about Category theory in the online encyclopedia. (Site not responding. Last check: 2007-10-29)
		Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them.
		Categorical logic is now a well-defined field based on type theory for intuitionistic logics, with application to the theory of functional programming and domain theory, all in a setting of a cartesian closed category as non-syntactic description of a lambda calculus.
		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.onlineencyclopedia.org /c/ca/category_theory.html (2963 words)

	Cokernel - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-29)
		In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f.
		The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0
		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.
	www.encyclopedia-online.info /Cokernel (433 words)

	PlanetMath: dual category (Site not responding. Last check: 2007-10-29)
		For example, a coproduct is a product on the opposite category; this can be seen by looking at the commutative diagram that completely specifies a coproduct, and noting that it is the same as the diagram specifying a product with the arrows reversed.
		cokernel is a kernel in the opposite category.
		This is version 5 of dual category, born on 2002-02-25, modified 2004-03-29.
	planetmath.org /encyclopedia/DualCategory.html (227 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)

	PlanetMath: kernel is an inverse limit
		This is exactly the universal condition for a kernel in an abelian category.
		This result can be extremely useful in proving exactness results: one shows that finite inverse and direct limits exist and are exact in a particular category, and one immediately obtains the fact that sums, products, kernels and cokernels are all exact.
		This is version 2 of kernel is an inverse limit, born on 2004-02-25, modified 2004-02-25.
	planetmath.org /encyclopedia/KernelIsAnInverseLimit.html (162 words)

	Category (mathematics) (Site not responding. Last check: 2007-10-29)
		The study of categories in their own right is known as category theory.
		A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set.
		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.worldhistory.com /wiki/C/Category-(mathematics).htm (1221 words)

	Kernel (category theory) (Site not responding. Last check: 2007-10-29)
		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.
	www.worldhistory.com /wiki/K/Kernel-(category-theory).htm (888 words)

	Category theory biography .ms (Site not responding. Last check: 2007-10-29)
		Categories, functors and natural transformations were introduced by Samuel Eilenberg and Saunders Mac Lane in 1945.
		Each category is presented in terms of its objects, its morphisms, and its composition of morphisms.
		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.
	www.biography.ms /Category_theory.html (2366 words)

	math lessons - Abelian category
		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 (926 words)

	Category theory : search word
		Thus, the roads on a map are analagous to the morphisms of the category.
		Category is called locally small iff for every two objects A and B there is a set Mor(''A'',''B'') of morphisms from A to B.
		Main articles universal property, limit (category theory) 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.
	www.searchword.org /ca/category-theory.html (2907 words)

	Cokernel (Site not responding. Last check: 2007-10-29)
		As with all universal constructions the cokernel it exists is unique up to a unique isomorphism.
		In the category of groups the cokernel a group homomorphism f : G → H is the quotient of H by the normal closure of the of f.
		In such a category the coequalizer of two morphisms f and g (if it exists) is just the of their difference:
	www.freeglossary.com /Cokernel_(category_theory) (802 words)

	Pre-Abelian category
		A pre-Abelian category is an additive category that has all kernels and cokernelss.
		exists (this is the kernel), as does the coequaliser (this is the cokernel).
		; similarly, their coequaliser is the cokernel of their difference.
	www.guajara.com /wiki/en/wikipedia/p/pr/pre_abelian_category.html (832 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.
		Since P is a generator, every S-module is the cokernel of a morphism between di* *rect sums of copies of P, so the counit is bijective in general.
	hopf.math.purdue.edu /Schwede/Morita.txt (4235 words)

	PlanetMath: abelian category
		See Also: supplemental axioms for an Abelian category
		This is version 8 of abelian category, born on 2002-04-22, modified 2004-05-05.
		(Category theory; homological algebra :: Abelian categories :: Exact categories, abelian categories)
	www.planetmath.org /encyclopedia/Cokernel.html (155 words)

	Category theory + physics - Physics Help and Math Help - Physics Forums
		I don't know why category theory always gets stick but other formalisms such as group theory or differential manifolds are learnt by physics students without a murmur.
		The first definition of any importance in category theory is that of a morphism ("map that respects the given structure").
		category theory also gives us a way to think about limits - we can describe ('infinite' in some sense usually) things as certain kinds of limits of 'finite' things, eg a vector space can be thought of as the limit of all its finite subspaces.
	www.physicsforums.com /showthread.php?threadid=79227 (3029 words)

	[No title]
		The theory is based on three standard exampl* es: the category* of topological spaces, the category of simplicial sets, and the ca* *tegory of chain complexes over a given ring.
		Definition 2.1.A model category is a category M equipped with three distin- guished classes of maps: the weak equivalences, the cofibrations, and the fibra* *tions.
		The category Mod- A has a model category structure where the weak equivalences are the maps inducing an isomorphism in h* *o- mology and the fibrations are the surjections.
	hopf.math.purdue.edu /Dugger-Shipley/kdeqDS.txt (10785 words)

	Kernel (category theory) - InfoSearchPoint.com (Site not responding. Last check: 2007-10-29)
		Intuitively, the kernel of the morphism f : A → B is the "most general" morphism k : K → A which, when composed with f, yields zero.
		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.
		When m is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel.
	www.infosearchpoint.com /display/Kernel_(categories) (822 words)

	The world's top category theory websites
		The category Med consisting of all medial magmas together with their homomorphisms.
		The category Ab consisting of all abelian groups together with their group homomorphisms.
		Any preordered set (P, ≤) forms a small category, where the objects are the members of P, and the morphisms are arrows pointing from x to y when x ≤ y.
	dirs.org /wiki-article-tab.cfm/category_theory (3216 words)

	Cokernel - TheBestLinks.com - Abstract algebra, Abelian group, Category theory, Quotient group, ... (Site not responding. Last check: 2007-10-29)
		Cokernel - TheBestLinks.com - Abstract algebra, Abelian group, Category theory, Quotient group,...
		Cokernel, Abstract algebra, Abelian group, Category theory, Quotient group...
		You can add this article to your own "watchlist" and receive e-mail notification about all changes in this page.
	www.thebestlinks.com /Cokernel.html (474 words)

	week99
		A zero object in a category is an object that is both initial and terminal.
		Besides its additive properties (zero object, binary coproducts, and cokernels), Hilb is also a monoidal category: we can multiply Hilbert space by tensoring them, and there is and a multiplicative identity, namely the complex numbers C. In fact, Hilb is a "ring category", as defined by Laplaza and Kelly.
		In quantum theory the inner product represents the amplitude to pass from v to w, while in category theory hom(x,y) is the set of ways to get from x to y.
	math.ucr.edu /home/baez/week99.html (2847 words)

	cokernel - OneLook Dictionary Search (Site not responding. Last check: 2007-10-29)
		We found 3 dictionaries with English definitions that include the word cokernel:
		Tip: Click on the first link on a line below to go directly to a page where "cokernel" is defined.
		Cokernel : Eric Weisstein's World of Mathematics [home, info]
	www.onelook.com /?w=cokernel (80 words)

	Cokernel (Site not responding. Last check: 2007-10-29)
		In mathematics, the cokernel of a homomorphism ''f : X → Y is the quotient of Y by the image of f.
		This notion is dual to the kernels of category theory, hence the name.
		Help build the largest human-edited directory on the web.
	www.serebella.com /encyclopedia/article-Cokernel.html (122 words)

	Cokernel category theory - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-10-29)
		Cokernel category theory - Term Explanation on IndexSuche.Com
		Books and Others to each Topic: "Cokernel (category theory)".
		A copy of the license is included in the section entitled
	www.indexsuche.com /Cokernel_(category_theory).html (59 words)

	Kernel (category theory) - ArtPolitic Encyclopedia of Politics : Information Portal
		Kernel (category theory) - ArtPolitic Encyclopedia of Politics : Information Portal
		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.
		While a category cannot have kernels unless it has zero morphisms, there is no guarantee that a category with zero morphisms must have kernels.
	www.artpolitic.org /infopedia/ke/Kernel_of_a_morphism.html (857 words)

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