Cokern

Cokern, short for "cokernel," is a concept in category theory, a branch of mathematics that provides a general framework for various mathematical structures and their relationships. The term "cokern" is often used in the context of abelian categories, which are categories where objects can be added together and where morphisms have kernels and cokernels.

In an abelian category, the cokernel of a morphism f: A → B is a morphism B → Cok(f) that satisfies certain universal properties. Intuitively, the cokernel captures the "coimage" of the morphism f, which is the quotient of B by the image of f. The cokernel is dual to the kernel, which captures the "image" of the morphism f.

The cokern of a morphism f: A → B can be constructed as the cokernel of the induced map A → Im(f), where Im(f) is the image of f. The cokern of f is then the quotient of B by Im(f). In the category of groups, for example, the cokernel of a group homomorphism f: G → H is the quotient group H/Im(f).

The concept of cokern is important in various areas of mathematics, including homological algebra, where it is used to define derived functors and cohomology theories. It is also used in algebraic geometry, where it is used to define sheaf cohomology and other cohomological invariants. In these contexts, the cokern provides a way to measure the "failure" of a functor to be exact, or the "obstruction" to the existence of certain algebraic structures.