The best-known abelian categories are categories of modules over various rings. And as modules, these objects are structured sets. Now, even though we’re willing to elide the difference between a hom-group and a hom-set, we would at least like to avoid talking about the objects as sets, and the morphisms as functions. So let’s try to focus on studying the morphisms and use them to understand the properties of the category. Luckily, in an abelian category we know a lot about morphisms.
For one thing, let’s consider an object 
and the set 
of all the arrows coming into 
. We can put the structure of a preorder 
on this set by saying 
if there is 
with 
. It’s straightforward to show that this relation is reflexive and transitive. Then, as usual we can symmetrize this relation to get an equivalence relation: 
if and only if 
and 
. When we pass to the equivalence classes, the preorder becomes a partial order, which we’ll also call 
. In particular, this includes all monomorphisms, which we know as subobjects.
Similarly, we can construct another preorder 
of all the arrows leading out of with 
if there is an 
with 
. We can also turn this into a partial order, which includes the partial order of all quotient objects of .
Now for every arrow 
from pick a kernel and for every 
to pick a cokernel. Then and 
live in , while 
and live in 
. And we have three equivalent statements:
Now the partial orders and can be considered as categories, as any partial order can be. And the maps 
and 
reverse these orders, so they are contravariant functors. So let’s flip the order on (
instead of 
) and instead consider them as covariant functors 
and 
.
The above equivalence now reads 
. That is, we have an adjunction. The identities satisfied by the unit and counit read, in this case:
September 18, 2007
Posted by John Armstrong |
Category theory |
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The Unapologetic Mathematician 