equaliser

Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if f : X → Y is an arbitrary morphism in C, then a kernel of f is an equaliser of f and the zero morphism from X to Y. In symbols: ker(f) = eq(f, 0XY) To be more explicit, the following universal property can be used. A kernel of f is any morphism k : See 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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Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in gener ...

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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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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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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 mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In s ...

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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

Read more here: » Kernel category theory: Encyclopedia II - Kernel category theory - Relation to other categorical concepts

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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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 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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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

Read more here: » Preadditive category: Encyclopedia II - Preadditive category - Kernels and cokernels