The Category of Representations is Abelian

We’ve been considering the category of representations of an algebra , and we’re just about done showing that is abelian.

First of all, the intertwiners between any two representations form a vector space, which is really an abelian group plus stuff. Since the composition of intertwiners is bilinear, this makes into an -category. Secondly, we can take direct sums of representations, which is a categorical biproduct. Thirdly, every intertwiner has a kernel and a cokernel.

The only thing we’re missing is that every monomorphism and every epimorphism be normal. That is, every monomorphism should actually be the kernel of some intertwiner, and every epimorphism should actually be the cokernel of some intertwiner. So, given representations and , let’s consider a monomorphic intertwiner .

As for linear maps, it’s straightforward to show that is monomorphic if and only if its kernel is trivial. Specifically, we can consider the inclusion and the zero map . It’s easy to see that , and so the left-cancellation property shows that , which is only possible if . So a monomorphism has a trivial kernel. Thus the underlying linear map is an isomorphism of onto the image . Then this subrepresentation is exactly the kernel of the quotient map . And so the monomorphism is the kernel of some map. The proof that any epimorphism is normal is dual.

And so we have established that the category of representations of the algebra is abelian. This allows us to bring in all the machinery of homological algebra, if we should so choose. In particular, we can talk about exact sequences, which can be useful from time to time.