Kernel category theory - Relationship to algebraic kernels

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

Read more here: » Kernel category theory: Encyclopedia II - Kernel category theory - Definition

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

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

Read more here: » Kernel category theory: Encyclopedia II - Kernel category theory - Examples