Cokernel

In mathematics, the cokernel of a homomorphism f: X→Y is the quotient of Y by the image of f. In a topological setting, one typically takes the closure of the image before passing to the quotient. For instance, if f: H₁→H₂ is a bounded linear operator between Hilbert spaces, then coker(f) is the quotient of H₂ by the closure of the range of f.

In the general framework of a category with zero morphisms, the cokernel of f : X→Y (if it exists) is the morphism g: Y→Z such that the composition gf is the zero map from X to Z and g is universal for this property, i.e., any h: Y→W such that hf = 0 can be obtained by composing g with a unique map from Z to W.

This notion is dual to the kernels of category theory, hence the name.