Zero objects, Kernels, and Cokernels
A zero object in a category is, simply put, both initial and terminal. Usually we’ll write for a zero object, but sometimes , or even in certain circumstances. While initial objects and terminal objects are both nice, zero objects are even nicer.
Since a zero object is terminal, there is a unique morphism in for each object . Since it’s initial, there’s a unique morphism in for each object . Now we can put these together: for any two objects and there is a unique morphism which factors through . Take the unique arrow from to , then the unique arrow from to . This picks out a special element of each hom set, just for having a zero object.
We saw that the trivial group in the category of groups is both initial and terminal, so it’s a zero object. If we’re just looking in the category we usually call the trivial abelian group , and it’s a zero object. The initial and terminal object in are different, so this category does not have a zero object.
It’s often useful to remedy this last case by considering the category of “pointed” sets. A pointed set is a pair where is a set and is any element of the set. A morphism of pointed sets is a function so that . The marked point has to go to the marked point. This gives us the category of pointed sets. It’s easily checked that the pointed set is both initial and terminal, so it is a zero object in .
If a category has a zero object then we have a special morphism in each hom set, as I noted above. If we have two morphisms between a given pair of objects we can ask about their equalizer and coequalizer. But now we have one for free! So given any arrow and the special zero arrow , we can ask about their equalizer and coequalizer. In this special case we call them the “kernel” and “cokernel” of , respectively. I’ll say more about the kernel, but you should also think about dualizing everything to talk about the cokernel.
Given a morphism its kernel is an morphism so that as morphisms from to . Also, given any other morphism with we have a unique morphism with h=k\circ g$. This is just the definition of equalizer over again. As with equalizers, the morphism is monic, so we can view as a subobject of and as the inclusion morphism.
Let’s look at this in the case of groups. If we have a group homomorphism the kernel will be group included into by the monomorphism — a subgroup. We also have that , so everything that starts out in gets sent to the identity element in . If we have any other homomorphism whose image gets sent to the identity in , then it factors through , so the image of lands in the image of . That is, picks out the whole subgroup of that gets sent to the identity in under the homomorphism . And that’s exactly what we called the kernel of a group homomorphism way back in February.
Often enough we aren’t really interested in all equalizers — just kernels will do. So, if a category has a zero object and if every morphism has a kernel, we say that the category “has kernels”. Dually, we say it “has cokernels”. Having a zero object and having equalizers clearly implies having kernels, but it’s possible to have kernels without having all equalizers.