The dual of an abelian category is again abelian, since the axioms are all preserved by the reversing of arrows. For example, the category of finite-dimensional vector spaces over a field is easily seen to be dual to itself, since we can just take linear duals of vector spaces. However, this is the only example where I know a 'concrete' description of both an abelian category and its dual, where by concrete I mean describing the category as a variant of, say, modules over a ring or sheaves on some space.


What are other examples of dual pairs of abelian categories which can both be described 'nicely'? For example, is there a concrete description of the dual of the category of abelian groups?



Duals of Abelian Categories
A few (mostly trivial) examples (may be related to references in Tim Porter's answer?):


First of all, the self-duality of the category of vector spaces can be enriched to other self-dualities:


* The category of finite-dimensional representations of a group is self-dual.
* The category of finite-dimensional representations of a Lie algebra is self-dual.


(Of course, now we can generalize this to Hopf algebras.)


Another simple class of examples comes from taking the dual of some object, and asking what additional structure is necessary to reconstruct the original object. For instance:


* Let $V$ be a vector space. Consider $V^\*$. It has a natural topology, in which open subspaces are orthogonal complements of finite-dimensional subspaces of $V$. (This is
essentially weak topology if $V$ is viewed as a discrete space.) This gives a duality
between vector spaces and certain class of topological vector spaces that can be easily described explicitly. (A fancier way to view this: vector spaces are 'ind-finite-dimensional', so their dual are 'pro-finite-dimensional'.)


This can be modified to duality on Tate vector spaces, if you happen to like this sort of things.


There are also some examples in (algebraic) geometry (that are probably not as straightforward as the others).


* Category of perverse sheaves is self-dual.
* Category of holonomic $D$-modules is self-dual (this is related to the previous comment
by the Riemann-Hilbert).


Finally, there is a (somewhat cheap) trick to start with a duality on the derived category, then take the abelian category sitting inside it, and realize that it is dual to its image. For instance:


* Look at the derived category of constructible sheaves (say, on a scheme); it has Verdier's duality. Then the abelian subcategories corresponding to dual perversities are dual to each other. For instance, the category of constructible sheaves is dual to perverse sheaves for a certain perversity.
* Look at the derived category of coherent sheaves on a scheme; it has Serre's duality. Then the abelian subcategories corresponding to dual perversities are dual to each other. For instance, the category of coherent sheaves is dual to perverse sheaves for a certain perversity.


