cokernelf

The term "cokernel" refers to a fundamental concept in abstract algebra, particularly within the study of modules and groups. In category theory and homological algebra, the cokernel of a morphism (or homomorphism) is a generalization of the quotient structure seen in groups, rings, and vector spaces. Given a morphism *f*: *A* → *B* between two objects in a category, the cokernel is an object *C* equipped with a morphism *g*: *B* → *C* such that the composition *g ∘ f* is the zero morphism, and *C* satisfies a universal property: for any other object *D* with a morphism *h*: *B* → *D* such that *h ∘ f* is zero, there exists a unique morphism *k*: *C* → *D* making the diagram commute.

In the context of abelian groups, rings, or modules over a ring, the cokernel of a homomorphism *f*: *A* → *B* is the quotient group (or module) *B*/*f(A)*, where *f(A)* denotes the image of *A* under *f*. This quotient captures the "remaining" structure after accounting for the image of *f*, analogous to how the kernel captures the "missing" structure in the domain. The cokernel is thus a dual concept to the kernel, which is the kernel of the dual morphism in the opposite category.

The cokernel plays a crucial role in defining exact sequences, which are central to homological algebra. An exact sequence is a sequence of morphisms where the image of each morphism equals the kernel of the next, and the cokernel appears in the right-hand terms of such sequences. This structure helps classify objects in algebraic systems and study their properties through tools like homology and cohomology. The concept extends naturally to other algebraic structures, including vector spaces and sheaves, where it retains its role as a quotient-like construction.