CATLOC - Categorical localisation: methods and foundations

Starting date
March 1, 2017
Duration (months)
24
Departments
Computer Science
Managers or local contacts
Angeleri Lidia

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.

Sponsors:

Funds: assigned and managed by the department

Project participants

Collaboratori esterni

Jan Stovicek
Charles University Prague
Ryo Takahashi
Nagoya University
Giuseppe Rosolini
Università di Genova
Silvana Bazzoni
Università di Padova Matematica
Maria Emilia Maietti
Università di Padova
Alberto Tonolo
Università di Padova Matematica
Research areas involved in the project
Matematica discreta e computazionale
Associative rings and algebras - -
Matematica discreta e computazionale
Mathematical logic and foundations - -
Matematica discreta e computazionale
Category theory; homological algebra - Homological algebra

Activities

Research facilities