Definability and interpretations in tensor-triangulated and additive categories
Lead Research Organisation:
University of Manchester
Department Name: Mathematics
Abstract
Techniques from additive model theory have been effective in solving problems in representation theory. Many of the additive and triangulated categories occurring in representation theory are equipped with a tensor structure. The project will develop an enriched additive model theory to take account of this, considering definable additive categories with a monoidal structure and interpretation functors between these which preserve that structure. Expected areas of application include support varieties, contramodules and (derived) categories of sheaves.
Organisations
People |
ORCID iD |
| Mike Prest (Primary Supervisor) | |
| Rose Wagstaffe (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509565/1 | 01/10/2016 | 30/09/2021 | |||
| 1916274 | Studentship | EP/N509565/1 | 01/10/2017 | 31/03/2021 | Rose Wagstaffe |
| Description | A 2-category anti-equivalence has been established between skeletally small abelian categories with additive exact symmetric monoidal structures and definable subcategories of certain monoidal finitely accessible categories which satisfy certain closure conditions with respect to the monoidal structure. This correspondence can be viewed as a monoidal analogue of the 2-category anti-equivalence between ABEX and DEF given in (Prest and Rajani, 2010) and sheds light on how the additive model theoretic machinery often used in representation theory interacts with well enough behaved monoidal structures. In the setting of rigidly-compactly generated tensor-triangulated categories, a correspondence has been found between definable tensor-ideals, Serre tensor-ideals of a corresponding functor category, cohomological ideals satisfying a closure condition and closed subsets of a new Ziegler-type topology. This is a tensor restriction of the Fundamental Correspondence established for compactly generated triangulated categories by Krause in 2002. When restricting this correspondence to triangulated definable tensor-ideals, the lattice of open subsets of the corresponding Ziegler-type topology is isomorphic to the lattice of smashing tensor-ideals of the triangulated category, thus proving that the lattice of smashing tensor-ideals forms a spatial frame. Furthermore, an internal tensor-duality between definable subcategories has been defined. |
| Exploitation Route | One could explore the applications of these result in the context of support varieties, contramodules and (derived) categories of sheaves. |
| Sectors | Other |
| URL | https://arxiv.org/abs/2010.12029 |