I was just wondering if someone could give a good motivation for why to go through with the somewhat convoluted category theoretical definition of the kernel of a morphism (paraphrasing Grillet, p. 602):
Let $A,B$ be objects of a category $\mathcal{C}$. Then $\kappa: K \rightarrow A$ ($K$ being some object of $\mathcal{C}$) is a kernel of $\alpha : A \rightarrow B$ if and only if $\alpha \circ \kappa = 0$, and every morphism $\varphi : X \rightarrow A$ ($X$ being some object of $\mathcal{C}$) such that $\varphi \circ \alpha = 0$ factors uniquely through $\kappa$.
Like, in what sense is "the set of elements $a \in A$ for which $\alpha(a) = 0$" not good enough? Why is it desirable to set up this convoluted affair with another object and another morphism, etc.?
Strikes me that there should probably be some good classical examples of this out there, where the 'original' definition fails and the more sophisticated category theoretical definition makes sense, but I fail to figure out the right terms to put into Google to find them.
Does anyone know of any?
What are the elements of an object in an arbitrary category ?
Say for instance you're looking at the category of sheaves of abelian groups over a space $X$. Then you have elements of $A(U)$ for every open subset $U$ of $X$, not "elements of $A$".
And this is still a pretty mild example (since we still have some notion of element, only we have to do it for every open subset), you can do worse. Some categories really don't have such a notion.
The definition via morphisms allows you to forget the way you built your objects (out of sets - when you do algebra for instance, you're not really interested in the sets, you're interested in the structure), and get to the essential property of the kernel : the place where things are $0$.
This is good because this property is what we really want in a kernel, and it allows us to generalize to arbitrary settings, even when you may not have a(n) (obvious) notion of elements