Triangulated categories arise in all areas of mathematics dealing with homological algebra. In representation theory, the focus is mostly on the derived category of modules over a ring; in algebraic geometry it is the derived category of (quasi)coherent sheaves that takes centre stage; and in homotopy theory it is the homotopy category of spectra. Each area has developed different approaches to deal with its own problems, but there are questions of a transversal character, which form the body of this project.
What is the nature of these problems? Roughly speaking, they have to do with the classification up to equivalence and the decomposition of triangulated categories. For derived categories of abelian categories, usually equivalences are parametrised by so-called tilting objects. Tilting, or more generally, silting theory provides us with some control of the existence and shape of derived equivalences, and with a way to study derived invariants. Alternatively, in order to understand a larger algebraic structure, one wants to decompose it into smaller pieces. It is important that these smaller pieces come with enough data that allow us to glue our understanding of the pieces to the larger structure.
Categorical localisations are fundamental to carry out such decompositions, and to allow the glueing. Among the localisation techniques in the setting of triangulated categories, recollements or, as known in algebraic geometry, Grothendieck's six functor formalism, play a central role. Recollements of derived module categories are often induced by ring-theoretic localisations such as universal localisations.
We study the interplay between different localisation techniques and we explore applications to relevant contexts.
Moreover, we address some computational and foundational issues raised by localisation techniques which are within the range of the homotopy type theory developed by Voevodsky and others.