# Better understanding of the pre sheaf kernel

I'm currently reading chapter II of Hartshorne's algebraic geometry and there defines kernel ,cokernel.....etc. everything is define on sheaf of Abelian groups.

we know that for a sheaf morphism $$\phi :\mathcal{F}-\mathcal{G}$$ kernel is $$\ker{\varphi}$$ for sheaf of abelian group or ring ...etc. But if $$\mathcal{F},\mathcal{G}$$ are sheaf of sets, then what is the equivalent condition of kernel .

• If the topological space $X$ you're working over, then sheaves of sets, groups, etc are just sets, groups, etc. What is the kernel for a map of sets? I don't know of a nice notion of kernel even in this setting. Commented Nov 14, 2019 at 12:59
• The kernel of $\phi$ is the equalizer of $\phi$ and $0$. Is that what you mean? See en.wikipedia.org/wiki/Equaliser_(mathematics) Commented Nov 14, 2019 at 13:59
If $$C$$ is an abelian category (i.e. if it has kernels, cokernels, and some additional properties), then the category of sheaves of elements of $$C$$ is itself an abelian category, by the constructions you mentioned. This means sheaves of abelian groups, rings, modules, etc. have well-defined kernels and cokernels. However, $$\text{Set}$$ is not an abelian category, and so we should not expect the category of sheaves of sets to be abelian, either.