Cokernels - how to explain or get a good intuition of what they are or might be When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over the place, whenever I see a homomorphism. BUT no-one really seems to mention cokernels until you get to commutative diagrams and category theory. And then they can easily just be "things which make the diagram work" with limited intuition or sense of useful reality. [maybe I exaggerate] So I am looking for good examples to illustrate what a cokernel is, extending to non-trivial examples [I was taught about the kernels of homomorphisms between non-abelian groups before anyone taught me about modules].

Let $\phi:A \to B$ be a homomorphism where $A$ and $B$ are your favorite algebraic objects. I think of $\ker \phi$ as measuring the extent to which the morphism $\phi$ is not injective. That is, the "larger" the kernel, the more the map $\phi$ differs from an injection.

In a similar vein, the cokernel $B /~\text{im}~ \phi$ measures the extent to which the map $\phi$ differs from a surjection. A "large" cokernel indicates that the map $\phi$ is far from being surjective.

As an example, consider the embeddings of vector spaces $f:\mathbb{R} \to \mathbb{R^2}$ and $g:\mathbb{R} \to \mathbb{R^3}$. The cokernel of $f$ is isomorphic to $\mathbb{R}$ and the cokernel of $g$ is isomorphic to $\mathbb{R}^2$. So, in a sense, $f$ is closer to being a surjection than $g$ is.