Definition, Meaning & Anagrams | English word COKERNEL

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Definitions of COKERNEL

1. (mathematics, category theory, of a morphism in a category with zero morphisms) The coequalizer of and the zero morphism from to , denoted .

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Examples of Using COKERNEL in a Sentence

• Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel.
• In an abelian category, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel.
• Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.
• On a compact manifold, it follows from the elliptic theory that P is a Fredholm operator: it has finite-dimensional kernel and cokernel.
• Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).
• In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism.
• More generally, if W is a linear subspace of a (possibly infinite dimensional) vector space V then the codimension of W in V is the dimension (possibly infinite) of the quotient space V/W, which is more abstractly known as the cokernel of the inclusion.
• Loosely speaking, one corrects for this effect by forming from the cokernel of D a vector bundle, called the obstruction bundle, and then realizing the GW invariant as the integral of the Euler class of the obstruction bundle.
• More abstractly, the codimension of a map is the dimension of the cokernel, while the relative dimension of a map is the dimension of the kernel.
• In mathematics, corank is complementary to the concept of the rank of a mathematical object, and may refer to the dimension of the left nullspace of a matrix, the dimension of the cokernel of a linear transformation of a vector space, or the number of elements of a matroid minus its rank.
• In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel.

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