Properties of Ab-Categories
There are a number of things we can say right off about the -categories we defined last time. As is common practice, we’ll blur the distinction between an abelian group and its underlying set.
First of all, any -category has zero morphisms. That is, there’s a special morphism between any two objects that when composed with any other morphism gives the special morphism in the appropriate hom-set. In fact, since each hom-set is an abelian group it has an additive identity . Then for any we have , which composition must send to . The zero morphisms are exactly the zero morphisms!
Given any object the hom-set is already an abelian group. But the composition puts the structure of a monoid onto this set as well, and the linearity condition says these two are compatible, making the endomorphism monoid into an endomorphism ring. In fact, every ring is an endomorphism ring. Way back when we first defined categories we noted that a category with one object was the exact same thing as a monoid. And a ring is just an abelian group with a compatible monoid structure on it. So an -category with a single object is the exact same thing as a ring! In fact, a lot of the study of -categories can be seen as extending ring theory from that special case to the more general one. Incidentally, you should see right off that when we consider rings and as categories like this, a ring homomorphism from to is the same thing as an -functor between the categories.
Remember when we talked about direct sums of modules over a given ring? Well the same thing happens here. We define the “biproduct” of the finite collection of objects to be an object along with two families of arrows:
satisfying the relations
From the same arguments as in our coverage of direct sums we see that a biproduct satisfies the universal properties of both a categorical product and coproduct, and conversely that a categorical product or coproduct implies the existence of the biproduct arrows. Note that we’re making no statement whatsoever that such a biproduct actually exists in our category, but when it does it’s both a product and a coproduct.
As a special case, we can consider the biproduct of an empty collection of objects. This will be both a product and a coproduct of an empty collection of objects, if it exists, and will thus be a zero object. Of course, it may or may not exist.
Even if there is no zero object in our category, we still have the above zero morphisms, and so we can still talk about kernels and cokernels. The kernel of a morphism is the equalizer , and its cokernel is the coequalizer . In fact, life is even better now that we’re enriched over : every equalizer is a kernel and every coequalizer is a cokernel. Indeed, and similarly for coequalizers. Again, we’re saying nothing about whether such kernels or cokernels actually exist.
Together, these facts say a lot about the behavior of limits in -categories. Biproducts tell us about finite products and coproducts, while kernels of morphisms tell us about all different equalizers. And then The Existence Theorem for Limits tells us that every finite limit can be constructed from finite products and equalizers, while every finite colimit can be constructed from finite coproducts and coequalizers. So if our -category has all biproducts, all kernels, and all cokernels, then it has all finite limits whatsoever!
Let’s add one more little property that will simplify our life. We know that kernels are monomorphisms, and that cokernels are epimorphisms. If we assume on top of having all biproducts, kernels, and cokernels that every monomorphism is actually the kernel of some arrow in our category, and that every epimorphism is actually the cokernel of some arrow, then we will call our -category an abelian category.
You should verify that given any ring the category of all left -modules satisfies all these properties, and thus is an abelian category. These are the abelian categories that started the whole theory of homological algebra, which is to a large extent the study of general abelian categories.