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Quotient space of a codomain of a linear map by the map's image
The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the
Cokernel
Theorem in homological algebra
zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c: ker ( f ) ⟶ ker a ⟶ ker b ⟶ ker c
Snake_lemma
Category
pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian
Pre-abelian_category
Category with direct sums and certain types of kernels and cokernels
in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example
Abelian_category
is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel. A quasi-abelian category is an exact category.[citation
Quasi-abelian_category
Complementary of a rank
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
Corank
Tool in homological algebra
theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that
Mapping cone (homological algebra)
Mapping_cone_(homological_algebra)
Mathematical category whose hom sets form Abelian groups
exist morphisms without kernels and/or cokernels. There is a convenient relationship between the kernel and cokernel and the abelian group structure on the
Preadditive_category
Mathematical function, in linear algebra
invariant of a linear transformation f : V → W {\textstyle f:V\to W} is the cokernel, which is defined as coker ( f ) := W / f ( V ) = W / im ( f ) . {\displaystyle
Linear_map
Generalization of the kernel of a homomorphism
The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa. As
Kernel_(category_theory)
Type of morphism
the kernel of some morphism, and an epimorphism is conormal if it is the cokernel of some morphism. A category C is binormal if it's both normal and conormal
Normal_morphism
Set of eigenvalues of a matrix
operator is semi-Fredholm if its range is closed and either its kernel or cokernel (or both) is finite-dimensional.) Example 1: λ = 0 ∈ σ e s s , 1 ( A )
Spectrum (functional analysis)
Spectrum_(functional_analysis)
Aspect of category theory
particularly simple. It is just the factor group Y / im(f – g). (This is the cokernel of the morphism f – g; see the next section). In the category of topological
Coequalizer
transformation from the kernel of some primary cohomology operation to the cokernel of another primary operation. They were introduced by J. Frank Adams (1960)
Secondary cohomology operation
Secondary_cohomology_operation
Vectors mapped to 0 by a linear map
\operatorname {rank} (A)+\operatorname {nullity} (A)=n.} The left null space, or cokernel, of a matrix A consists of all column vectors x such that xTA = 0T, where
Kernel_(linear_algebra)
Topics referred to by the same term
linear map, the dimension of the map's kernel minus the dimension of its cokernel Index of a matrix Index of a real quadratic form Index, the winding number
Index
Elements taken to zero by a homomorphism
homomorphism has a kernel and cokernel, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel. Kernels of
Kernel_(algebra)
defined as the equalizer ( I m , m ) {\displaystyle (Im,m)} of the so-called cokernel pair ( Y ⊔ X Y , i 1 , i 2 ) {\displaystyle (Y\sqcup _{X}Y,i_{1},i_{2})}
Image_(category_theory)
Part of Fredholm theories in integral equations
kernel ker T {\displaystyle \ker T} and finite-dimensional (algebraic) cokernel coker T = Y / ran T {\displaystyle \operatorname {coker} T=Y/\operatorname
Fredholm_operator
General theory of mathematical structures
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Category_theory
In mathematics, invertible homomorphism
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Isomorphism
Mathematical result in differential geometry
the kernel of D (solutions of Df = 0), and the (finite) dimension of the cokernel of D (the constraints on the right-hand-side of an inhomogeneous equation
Atiyah–Singer_index_theorem
p=p} . Elementary considerations show that every idempotent then has a cokernel. The pseudo-abelian condition is stronger than preadditivity, but it is
Pseudo-abelian_category
constructed as a localization of A by the class of morphisms whose kernel and cokernel are both in B. An isogeny from an abelian variety A to another one B is
Localization_of_a_category
Concept in mathematics
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Tensor–hom_adjunction
Eigenvector with vanishing eigenvalue
circle. The kernel of an operator consists of left zero modes, and the cokernel consists of the right zero modes. Vaughn, Michael T. (2008). Introduction
Zero_mode
Generalization of vector bundles
they are closed under operations such as taking kernels, images, and cokernels. The quasi-coherent sheaves are a generalization of coherent sheaves and
Coherent_sheaf
In algebra, a module over a ring
can be visualized as an (infinite) matrix with entries in R and M as its cokernel. A free presentation always exists: any module is a quotient of a free
Free_presentation
Sequence of homomorphisms such that each kernel equals the preceding image
notion of an exact sequence makes sense in any category with kernels and cokernels, and more specially in abelian categories, where it is widely used. To
Exact_sequence
Collection of objects and morphisms
additive category. If all morphisms have a kernel and a cokernel, and all epimorphisms are cokernels and all monomorphisms are kernels, then we speak of an
Category_(mathematics)
Whether a manifold which is a homotopy sphere is a sphere
r ( J n ) ≡ c o k e r n e l ( J n ) {\displaystyle coker(J_{n})\equiv cokernel(J_{n})} . The remaining term in the short exact sequence, namely the group
Generalized Poincaré conjecture
Generalized_Poincaré_conjecture
Applications of category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Applied_category_theory
Mathematical construction used in homotopy theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Simplicial_set
Category enriched over the category of simplicial sets
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Simplicially enriched category
Simplicially_enriched_category
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Refinement_(category_theory)
Then Ω − 1 ( M ) {\displaystyle \Omega ^{-1}(M)} is defined to be the cokernel of i. A case of particular interest is when the ring R is a group algebra
Stable_module_category
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Tetracategory
Map (arrow) between two objects of a category
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Morphism
Category whose objects are abelian groups and whose morphisms are group homomorphisms
homomorphism i : K → A {\displaystyle i:K\to A} . The same is true for cokernels; the cokernel of f is the quotient group C = B / f ( A ) {\displaystyle C=B/f(A)}
Category_of_abelian_groups
Embedding of categories into functor categories
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Yoneda_lemma
Vector space consisting of affine subsets
(the nullity of T) plus the dimension of the image (the rank of T). The cokernel of a linear operator T : V → W is defined to be the quotient space W/im(T)
Quotient space (linear algebra)
Quotient_space_(linear_algebra)
Relationship between two functors abstracting many common constructions
that the cokernel functors for abelian groups, vector spaces and modules are left adjoints. Coproducts, pushouts, coequalizers, and cokernels are all examples
Adjoint_functors
Theorem in category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Lawvere's_fixed-point_theorem
Injective homomorphism
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Monomorphism
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
N-group_(category_theory)
In number theory, measure of non-unique factorization
homomorphism; its kernel is the group of units of R {\displaystyle R} , and its cokernel is the ideal class group of R {\displaystyle R} . The failure of these
Ideal_class_group
Type of category in category theory
category is an additive category in which every morphism has a kernel and a cokernel. An abelian category is a pre-abelian category such that every monomorphism
Additive_category
Topics referred to by the same term
category in which every monomorphism is a kernel and every epimorphism is a cokernel Abelian and Tauberian theorems, in real analysis, used in the summation
Abelian
Sequence of terms related to the first step of a spectral sequence
of H 1(A) by the first subgroup in the filtration, and hence it is the cokernel of the edge map from E21,0. This yields a short exact sequence 0 → E21
Five-term_exact_sequence
Most general completion of a commutative square given two morphisms with same codomain
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Pullback_(category_theory)
About direct sums and exact sequences
that: C ≅ B/ker r ≅ B/q(A) (i.e., C isomorphic to the coimage of r or cokernel of q) to: B = q(A) ⊕ u(C) ≅ A ⊕ C where the first isomorphism theorem is
Splitting_lemma
Special objects used in (mathematical) category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Initial_and_terminal_objects
Mathematical group
G → Aut(G). The kernel of the conjugation map is the center, while the cokernel is the outer automorphism group (and the image is the inner automorphism
Outer_automorphism_group
Difference between two dimensions
fiber. 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.
Relative_dimension
Overview of and topical guide to category theory
theory)/fiber product Inverse limit Pro-finite group Colimit Coproduct Coequalizer Cokernel Pushout (category theory) Direct limit Biproduct Direct sum Preadditive
Outline_of_category_theory
Set of elements that commute with every element of a group
Inn(G). By the first isomorphism theorem we get, G/Z(G) ≃ Inn(G). The cokernel of this map is the group Out(G) of outer automorphisms, and these form
Center_(group_theory)
Branch of mathematics
restricts oneself to finitely generated modules. However, every module is a cokernel of a homomorphism of free modules. Modules over the integers can be identified
Linear_algebra
Relation of categories in category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Isomorphism_of_categories
Concept in category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Monoidal_functor
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Fundamental_groupoid
Tool to track locally defined data attached to the open sets of a topological space
projective limits commute with projective limits. On the other hand, the cokernel is not always a sheaf because inductive limits do not necessarily commute
Sheaf_(mathematics)
Homological algebra is the study of homological functors
complexes, which can be studied through their homology and cohomology. Cokernel Exact sequence Chain complex Differential module Five lemma Short five
List of homological algebra topics
List_of_homological_algebra_topics
German scientist and mathematician (1884–1941)
of such an operator, giving an example of an operator whose kernel and cokernel have different finite dimension and providing a formula for the difference
Fritz_Noether
Most general completion of a commutative square given two morphisms with same domain
B ⊔C A. In an abelian category all pushouts exist, and they preserve cokernels in the following sense: if (P, i1, i2) is the pushout of f : Z → X and
Pushout_(category_theory)
categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence. An exact
Exact_category
Aspect of category theory in mathematics
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Rig_category
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Diagonal_functor
Matrix decomposition
singular value 0 {\displaystyle 0} comprise all unit vectors in the cokernel and kernel, respectively, of M {\displaystyle \mathbf {M} } . By the
Singular_value_decomposition
transpose Tr(M) of a left Λ-module M is a right Λ-module defined to be the cokernel of the map Q* → P*, where P → Q → M → 0 is a minimal projective presentation
Artin_algebra
Abstract mathematics relationship
equalizers, products and coproducts among others. Applying it to kernels and cokernels, we see that the equivalence F is an exact functor. C is a cartesian closed
Equivalence_of_categories
Concept in homotopy theory
{\displaystyle i:C_{\bullet }\to D_{\bullet }} which are degreewise monic and the cokernel complex Coker ( i ) ∙ {\displaystyle {\text{Coker}}(i)_{\bullet }} is a
Cofibration
Characterizing property of mathematical constructions
compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces. Before
Universal_property
Topological structure in number theory
follows. A quasi-isomorphism of modules is a homomorphism whose kernel and cokernel are both finite groups, in other words modules with support either empty
Iwasawa_algebra
Category theory concept
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Overcategory
Mapping between categories
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Functor
Functors which are surjective and injective on hom-sets
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Full_and_faithful_functors
can be block diagonalized with blocks corresponding to the kernel and cokernel of A s {\displaystyle A_{s}} . The Drazin inverse in the same basis is
Drazin_inverse
Category in which all small limits exist
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Complete_category
Generalization of category
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
2-category
Theorem in algebraic geometry that builds a homotopy equivalent affine variety
( 1 ) {\displaystyle {\mathcal {O}}_{X}(1)} and the second map is the cokernel. Jouanolou then asserts that W is a torsor for E = Hom ( F , O X ( −
Jouanolou's_trick
Branch of mathematics
notion of an exact sequence makes sense in any category with kernels and cokernels. The most common type of exact sequence is the short exact sequence. This
Homological_algebra
Abelian categories, while abstractly defined, are in fact concrete categories of modules
of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod. Such an equivalence
Mitchell's_embedding_theorem
Category admitting tensor products
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Monoidal_category
Collection of maps which give the same result
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Commutative_diagram
closure. CLT – central limit theorem. cod, codom – codomain. cok, coker – cokernel. colsp – column space of a matrix. conv – convex hull of a set. Cor – corollary
List of mathematical abbreviations
List_of_mathematical_abbreviations
Mathematical concept
categories. Coequalizers are colimits of a parallel pair of morphisms. Cokernels are coequalizers of a morphism and a parallel zero morphism. Pushouts
Limit_(category_theory)
Category theory constructs
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Kan_extension
Commutative group (mathematics)
generator of the kernel. Then, the abelian group is isomorphic to the cokernel of linear map defined by M. Conversely every integer matrix defines a finitely
Abelian_group
Mathematical operation on vector spaces
\qquad a_{ij}\in R,} the tensor product can be computed as the following cokernel: M ⊗ R N = coker ( N J → N I ) {\displaystyle M\otimes _{R}N=\operatorname
Tensor_product
Type of category in category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Cartesian_closed_category
module whose finitely generated submodules are finitely presented. cokernel The cokernel of a module homomorphism is the codomain quotiented by the image
Glossary_of_module_theory
Mathematical category
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Topos
Process in digital electronics and integrated circuit design
Retrieved 2018-01-15; Rutenbar, Rob A. Multi-level minimization, Part II: Cube/Cokernel Extract (PDF) (lecture slides). Carnegie Mellon University (CMU). Lecture
Logic_optimization
Set of arguments where two or more functions have the same value
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Equaliser_(mathematics)
Central object of study in category theory
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Natural_transformation
Topics referred to by the same term
case striking down the death sentence of a defendant convicted of rape Cokernel, written coker {\displaystyle \operatorname {coker} } , a concept in mathematics
Coker
Difference between the dimensions of mathematical object and a sub-object
infinite) of the quotient space V/W, which is more abstractly known as the cokernel of the inclusion. For finite-dimensional vector spaces, this agrees with
Codimension
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Lift_(mathematics)
Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations
Elementary_topos
Mathematical category with weak equivalences, fibrations and cofibrations
cofibrations are maps that are monomorphisms in each degree with projective cokernel; and fibrations are maps that are epimorphisms in each nonzero degree or
Model_category
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