Cokernel

The cokernel of a morphism is its codomain modulo its image. It measures which parts of the codomain the morphism fails to reach.

Image (mathematics): The image of a function is the set of outputs it attains; for a morphism, it is a subobject of its codomain. The cokernel divides the codomain by precisely this subobject.
Universal property: A universal property characterizes an object through a uniquely specified pattern of maps to or from other objects. It characterizes the cokernel without depending on a particular quotient construction.
First isomorphism theorem: The first isomorphism theorem identifies a homomorphism’s image with its domain modulo its kernel. Together with a cokernel, it separates the map’s failure of injectivity from its failure of surjectivity.
Kernel: The kernel of a morphism is the part of its domain sent to zero. It detects non-injectivity, whereas the cokernel detects non-surjectivity.
Quotient group: A quotient group groups elements of a group into cosets of a normal subgroup. In groups, the cokernel is the quotient of the codomain by the image subgroup.
Abelian category: An abelian category is an additive category with kernels and cokernels, satisfying exactness axioms. Its axioms ensure cokernels behave coherently alongside kernels and images.
Homology group: A homology group measures cycles modulo boundaries in a chain complex. Homology is itself a quotient, paralleling the image-quotient construction defining cokernels.
Image (category theory): The image of a morphism is the subobject through which it factors as an epimorphism followed by a monomorphism. The image records what is reached; the cokernel records what remains after quotienting by it.
Kernel (algebra): The kernel of a homomorphism is the subobject of its domain mapped to zero. Kernel measures failure of injectivity, while cokernel measures failure of surjectivity.
Coimage: The coimage of a morphism is its domain modulo its kernel. The canonical map from coimage to image helps relate a morphism’s two quotient constructions.

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