
Review of Short Phrases and Links 
This Review contains major "Cokernel" related terms, short phrases and links grouped together in the form of Encyclopedia article.
Definitions
 A cokernel is a kernel in the opposite category.
 A cokernel is always an epimorphism.
 This cokernel is dual to the kernels of category theory, hence the name.
 The dual concept to that of kernel is that of cokernel.
 The cokernel of any morphism is a normal epimorphism.
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 AB1) Every morphism has a kernel and a cokernel.
 It follows in particular that every cokernel is an epimorphism.
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 Note that the only portion of this diagram that depends on the cokernel condition is the object C 7 and the final pair of morphisms.
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 A Fredholm operator is a bounded linear operator between two Banach spaces whose kernel and cokernel are finitedimensional and whose range is closed.
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 Embedding A into some injective object I 0, the cokernel of this map into some injective I 1 etc., one constructs an injective resolution of A, i.e.
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 Even experienced students may have diﬃculty absorbing the abstract deﬁnitions of kernel, cokernel, product, coproduct, direct and inverse limit.
 Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its cokernel has dimension 1.
 The dimension of the cokernel and the dimension of the image (the rank) add up to the dimension of the target space.
 That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
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 One can define the cokernel in the general framework of category theory.
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 An additive category is preabelian if every morphism has both a kernel and a cokernel.
 Specifically, e is the cokernel of its own kernel: e = coker(ker e).
 This is the dual notion to the kernel: just as the kernel is a sub space of the domain, the cokernel is a quotient space of the target.
 In fact, life is even better now that we're enriched over: every equalizer is a kernel and every coequalizer is a cokernel.
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 Dually, it is said to have cokernels if every morphism has a cokernel.
 This fact recalls the usual alternative characterization of isomorphisms in an abelian category as those maps whose kernel and cokernel both vanish.
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 That is, every monomorphism m can be written as the cokernel of some morphism.
 Like all coequalizer s, the cokernel q: Y → Q is necessarily an epimorphism.
 It follows in particular that every cokernel In mathematics, the cokernel of a morphism f: X → Y (e.g.
 In abstract algebra, the cokernel of a homomorphism f: X → Y is the quotient of Y by the image of f.
 The cokernel itself is the set of all images of One(s3) (it is a normal subgroup in the group of all images under rev).
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 Examples 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 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.
 This category is not abelian, however, since for example the map has no cokernel (which would be if it existed).
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 Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.
 On the other hand, every epimorphism in the category of groups is normal (since it is the cokernel of its own kernel), so this category is conormal.
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 Conversely an epimorphism is called normal (or conormal) if it is the cokernel of some morphism.
 That is, every epimorphism e: A → B can be written as the cokernel of some morphism.
 Like all coequalizers, the cokernel q: Y → Q is necessarily an epimorphism.
 In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism.
 Dually, we can also define the cokernel of a morphism,, to be the coequalizer of and.
 Colimits and diagonal functors Coproduct s, fibred coproducts, coequalizer s, and cokernel s are all examples of the categorical notion of a colimit.
 In mathematics, specifically in category theory, a preAbelian category is an additive category that has all kernels and cokernel s.
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 The kernel of a morphism is the equalizer, and its cokernel is the coequalizer.
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 If a category has zero morphisms, then one can define the notions of kernel and cokernel for any morphism in that category.
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 In an abelian category, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel.
 In such a category, the coequalizer of two morphisms f and g (if it exists) is just the cokernel of their difference: coeq(f, g) = coker(g  f).
 In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference: coeq(f, g) = coker(g  f).
 The converse assertion is false, in general; however, when morphisms in have kernels, then every normal epimorphism is a cokernel.
 A monomorphism is normal if it is the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism.
 Special cases A preAbelian category is an additive category in which every morphism has a kernel and a cokernel.
 In category theory: a normal morphism is a morphism that arises as the kernel or cokernel of some other morphisms.
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Cokernel
 Kernels and cokernels Because the homsets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense.
 If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an abelian category.
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 If all morphisms have a kernel and a cokernel, and all epimorphism are cokernels and all monomorphisms are kernels, then we speak of an abelian category.
Categories
 Science > Mathematics > Algebra > Morphism
 Kernel
 Knowledge > Categorization > Category Theory > Coequalizer
 Mathematics > Algebra > Homomorphism > Epimorphism
 Knowledge > Categorization > Category Theory > Additive Category

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