Certain torsion pairs in a triangulated category T, known as t-structures,
give rise to abelian subcategories (hearts) which are relevant to the study
of the structure of T. In the context of a compactly generated triangulated
category (for example, the derived category of a small dg category), we give
a description of which hearts are Grothendieck abelian categories. In particular,
we show that compactly generated t-structures have Grothendieck hearts. Moreover,
if time permits, we will discuss the relation between these results and
the development of a derived Morita theory (via cotilting theory) for
Grothendieck abelian categories.
This talk is based on joint work with Lidia Angeleri Hügel and Frederik Marks.