Is functional analysis a special case of tilting theory?

Dr Sven-Ake Wegner, Teesside University. Part of the Algebra seminar series.

The objects of functional analysis together with the corresponding morphisms don't form abelian categories. Classical examples, e.g., the category of Banach spaces, satisfy almost all axioms of an abelian category but the canonical morphism between the cokernel of the kernel and the kernel of the cokernel of a given map fails in general to be an isomorphism.  In 2003, Bondal and van den Bergh showed that there is a correspondence between (co-)tilting torsion pairs and so-called quasiabelian categories. Indeed, every quasiabelian category, in particular the category of Banach spaces, is derived equivalent to the (abelian) heart of the canonical t-structure on its derived category. In this talk we discuss examples of categories appearing in functional analysis that are not quasiabelian but which carry a natural exact structure. The derived category is thus defined. We are interested in an extension of the aforementioned equivalence as this could be a step towards a successful categorification of certain analytic problems.