cokernel

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). To be explicit, if f : X → Y is a homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then K is a subalgebra of X and the inclusion homomorphism from KSee also:

Kernel category theory, Kernel category theory - Definition, Kernel category theory - Examples, Kernel category theory - Relation to other categorical concepts, Kernel category theory - Relationship to algebraic kernels

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Let U : D → C be a functor from a category D to a category C, and let X be an object of C. A universal morphism from X to U consists of a pair (A, φ) where A is an object of D and φ : X → U(A) is a morphism in C, such that the following universal property is satisfied: Whenever Y is an object of D and f : X → U(Y) is a morphism in C, then there exists a unique morphism g : A → < ...

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Universal property, Universal property - Formal definition, Universal property - Properties, Universal property - Existence and uniqueness, Universal property - Equivalent formulations, Universal property - Relation to adjoint functors, Universal property - Examples, Universal property - Tensor algebras, Universal property - Kernels, Universal property - Limits and colimits, Universal property - What is it good for?, Universal property - History

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Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. 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. Unlike with products and coproducts, the kernel and cokernel ...

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Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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Recall that a functor F: C → D between preadditive categories is additive if it is an Abelian group homomorphism on each hom-set in C. But if the categories are additive, then an additive functor can also be characterised as any functor that preserves biproduct diagrams. That is, if B is a biproduct of A1,...,An in C with projection morphisms pj and injection morphisms ij, then F(B) sho ...

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Additive category, Additive category - Examples, Additive category - Elementary properties, Additive category - Additive functors, Additive category - Special cases, Additive category - Sources

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The original example of an additive category is the category Ab of Abelian groups with group homomorphisms. Ab is preadditive because it is a closed monoidal category, and the biproduct in Ab is the finite direct sum. Other common examples: The category of (left) modules over a ring R, in particular: the category of vector spaces over a field K. The algebra of matrices over a ring, thought of as a category as described below. These will give you an idea of what to think of; for more examples, f ...

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Additive category, Additive category - Examples, Additive category - Elementary properties, Additive category - Additive functors, Additive category - Special cases, Additive category - Sources

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A morphism f : a → b is called a monomorphism (or monic) if fg1 = fg2 implies g1 = g2 for all morphisms g1, g2 : x → a. an epimorphism (or epic) if g1f = g2f implies g1 = g2 for all morphisms g1, g2 : b → x.See also:

Category mathematics, Category mathematics - Definition, Category mathematics - Examples, Category mathematics - Types of morphisms, Category mathematics - Types of categories

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Given any pair A, B of objects in an abelian category, there is a special zero morphism from A to B. This can be defined as the zero element of the hom-set Hom(A,B), since this is an abelian group. Alternatively, it can be defined as the unique composition A → 0 → B, where 0 is the zero object of the abelian category. In an abelian category, every morphism f can be written as the composition of an epimorphism followed by a monomorphism. This epimorphism is called the coimage of f, whil ...

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Abelian category, Abelian category - Definitions, Abelian category - Examples, Abelian category - Elementary properties, Abelian category - Related concepts, Abelian category - History

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The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. (Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. In contrast, the category of all groups is not closed.) See medial category. Other common examples: The category of (left) modules over a ring R, in particular: the category of vector spaces over a field K. The algebra of m ...

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Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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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, F is additive iff, given any objects A and B of C, the function F: Hom(A,B) → Hom(F(A),F(B)) is a group homomorphism. Most fun ...

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Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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The index theorem has been proved, and reproved, as a general statement. Atiyah-Singer comment that the initial proof was based on that of the Hirzebruch-Riemann-Roch theorem (1954), and involved cobordism theory. The next approach was to mimic, in some sense, the Grothendieck-Riemann-Roch theorem. That is, to find a relative statement (this being a highly fashionable point of view in the 1960s), for which functoriality carried the main burden. This they did, by means of a 'pushforward' operation i! on elliptic operators wh ...

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Atiyah–Singer index theorem, Atiyah–Singer index theorem - An example on the circle, Atiyah–Singer index theorem - More formal statement, Atiyah–Singer index theorem - History, Atiyah–Singer index theorem - Proof techniques, Atiyah–Singer index theorem - Further developments

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The theorem came at the end of more than 100 years' development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. The index problem may have been posed in generality first in the late 1950s by Israel Gelfand. He noticed the homotopy invariance of the index, and asked for a formula by means of topological invariants. Special cases were worked out, by Soviet mathematicians. Given the examples from Hodge theory, Cauchy-Riemann operators in several variables, and the topologists' work on the Riemann-Roch Theorem at the time, the re ...

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Atiyah–Singer index theorem, Atiyah–Singer index theorem - An example on the circle, Atiyah–Singer index theorem - More formal statement, Atiyah–Singer index theorem - History, Atiyah–Singer index theorem - Proof techniques, Atiyah–Singer index theorem - Further developments

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We start with a compact smooth manifold M (without boundary), a vector bundle, "E" on M and an elliptic operator D on M. Here "D" is a differential operator acting on smooth sections of the vector bundle. The property of being elliptic is expressed by a symbol s that can be seen as coming from the coefficients of the highest order part of D. The symbol s is itself defined on a vector bundle, the cotangent bundle or phase space. "s" assigns to every point of the cotangent bundle a ...

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Atiyah–Singer index theorem, Atiyah–Singer index theorem - An example on the circle, Atiyah–Singer index theorem - More formal statement, Atiyah–Singer index theorem - History, Atiyah–Singer index theorem - Proof techniques, Atiyah–Singer index theorem - Further developments

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Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in additive categories can be characterised by the following biproduct condition: The object B is a biproduct of the objects A1,...,An iff there are projection morphisms pj: B → Aj and injection morphisms ij: Aj → B, such that (See also:

Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of R-modules for a commutative ring R. Homotopy theory in this context is homological algebra. Homology can then be viewed as a type of homotopy, allowing genera ...

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Model category, Model category - Motivation, Model category - Formal definition, Model category - Example: Topological spaces, Model category - Example: Chain complexes of R-modules, Model category - Some constructions, Model category - Homotopy and the homotopy category

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