# When are kernels (resp. cokernels) finite limits (resp. finite colimits)?

Under what conditions, in what contexts, are kernels (resp. cokernels) finite limits (resp. finite colimits)?

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...always? ${}{}$ – Qiaochu Yuan Oct 29 '12 at 21:59
Assuming you have the right definition of kernel and cokernel, of course, to say nothing of the assumptions needed so that such things even make sense... – Zhen Lin Oct 29 '12 at 22:11

I misunderstood the question slightly (I thought you were asking about equalizers and coequalizers). The most general definition of a kernel or cokernel that I'm aware of takes place in a category with zero morphisms. This includes categories with zero objects as well as $\text{Ab}$-enriched categories such as abelian categories.
In such a category, the kernel (resp. cokernel) of a map $f : a \to b$ may be defined as the equalizer (resp. coequalizer) of $f$ and the zero morphism $0 : a \to b$. In particular, kernels are a finite limit and cokernels are a finite colimit. In an $\text{Ab}$-enriched category, the equalizer of an arbitrary pair of parallel morphisms $f, g : a \to b$ is the kernel of $f - g$, so being able to compute kernels (resp. cokernels) is equivalent to being able to compute equalizers (resp. coequalizers).
A more general definition replaces the notion of kernel and cokernel with the notion of kernel pair and cokernel pair. This is a generalization of the equivalence relation on a set $X$ induced by a map $f : X \to Y$ and is a pullback, so in particular a finite limit. Cokernel pairs are dually given by pushouts, so in particular are finite colimits.