I will introduce various recollements between functor categories (a recollement can be thought of as a "short exact sequence of abelian categories"), including a "recipe" for constructing them. I will focus on two of these recollements and explain why they have the same kernel.
The Auslander-Gruson-Jensen duality is a duality between two functor categories which arise from representation theory. It corresponds to elementary duality of pp formulas, a concept from the model theory of modules.
In this talk, I will begin by explaining the Auslander-Gruson-Jensen duality. I will go on to show that this duality can be extended to a recollement whose kernel is the category of pure exact sequences. I will show that another recollement, which instead deals with contravariant functors, has the same kernel. I will then prove that there is a structural reason for this relationship, in the form of a commutative diagram. This ultimately explains why the term "pure exact sequence" has two equivalent definitions: One in terms of tensor products and the other in terms of hom-sets.