# Triangulated Categories and Applications (16w5040)

Arriving in Banff, Alberta Sunday, June 19 and departing Friday June 24, 2016

## Organizers

J. Daniel Christensen (University of Western Ontario)

Ivo Dell'Ambrogio (Universite de Lille)

Amnon Neeman (Australian National University)

Paul Balmer (University of California Los Angeles)

## Objectives

The goal of this workshop is to gather researchers of various backgrounds who use triangulated categories in their respective domains. By encouraging them to exchange their different methods, points of view, and conjectures, we hope to stimulate new avenues of research and collaboration. PhD students and postdocs, in particular, will be able to use triangulated categories as a bridge towards new territories.

Ideally, young researchers will be asked to present the most recent advances, while more experienced colleagues will provide overviews of the uses of triangulated categories in their fields. This formula was well received by the participants of the first edition.

The theory of triangulated categories is relatively simple and elegant. This makes it a most convenient language for formulating results and techniques, and may explain its ubiquity. While of course not everything can be done at the triangular level, what can be accomplished has aesthetic appeal as well as tremendous potential in applications. To illustrate the versatile nature of triangulated categories, let us mention below a few triangular subjects at the center of current research, together with their multidisciplinary applications.

Originating in stable homotopy, where spectra represent generalized cohomology theories (Brown, Adams, Bousfield), the abstract ``Brown Representability Theorems" (Neeman, Krause) were exported to other subjects, from Grothendieck duality to the theory of motives. These techniques encourage the study of ``large'' triangulated categories, which still needs to be undertaken in several of the areas of expertise that will be represented at the workshop.

Verdier localization and Bousfield localization were born in algebraic geometry and topology respectively - they are the main tool for constructing new triangulated categories out of old ones, making them a staple of the theory. The closely related notion of Rickard idempotents is invaluable in modular representation theory of finite groups. Recollements and t-structures were born in geometry (Beilinson, Bernstein, Deligne) and were first applied to the study of perverse sheaves. They are now also extensively used in representation theory and noncommutative algebra. It can also be interesting to classify them (Alonso, Bridgeland, Jeremias, Saorin, Stovicek). The related notion of weight structures, a.k.a. co-t-structures (Bondarko, Pauksztello), has been developed and successfully applied in motivic theory for instance. In that world, the slice filtrations (Voevodsky, Levine) have played an important role. The neighboring formalism of Grothendieck's ``six operations", with its origins in etale cohomology, has now also branched to several fields, including the motivic world (Ayoub).

Putting those ideas at work more systematically throughout mathematics, for instance in representation theory, seems to hold a great potential for new results and interesting connections.

In several domains, one can find a central result taking the form of the classification of thick subcategories in some triangulated category of interest. Starting in topology with the classification of the thick subcategories of finite spectra (Hopkins, Smith, Devinatz), the idea has been exported to the derived category of a commutative ring (Hopkins, Neeman) and that of a scheme (Thomason), as well as to the stable module category of finite groups (Benson, Carlson, Rickard) and finite group schemes (Friedlander, Pevtsova). In order to distill the underlying ideas, Balmer has developed an axiomatic theory of tensor triangulated categories which allows one to extend algebro-geometric thinking to other fields. Another approach, starting from a commutative ring acting on a large triangulated category (Benson, Iyengar and Krause), has also found applications - notably the extension of the Benson-Carlson-Rickard classification to the large stable module category. A unifying relative version of these ideas (Stevenson) can be applied to new classes of examples such as singularity categories. This is a growing subject with several open questions and very exciting ongoing developments, for instance in equivariant stable homotopy theory. Such results (in progress at present) will be ripe for distribution by the time of the meeting. The equivariant stable homotopy category being in some sense the initial equivariant triangulated category, the results will undoubtedly have repercussions in many fields - from representation theory to motivic stable homotopy theory.

A recent generalization of Bousfield localization, analogous to the etale topology generalizing the Zariski topology, is the theory of separable extensions of triangulated categories, as developed by Balmer and his collaborators since the last meeting. For the derived categories of algebraic geometry, separable extensions describe etale extensions of schemes, and recover the etale topology (Neeman). Separable extensions also cover restrictions to subgroups in many equivariant triangulated categories - from KK-theory to equivariant stable homotopy (Dell'Ambrogio, Sanders). These techniques have been combined with descent theory, and in representation theory the combination led to new calculations of endotrivial modules (Balmer, Carlson, Thevenaz). It remains to explore this further, both in representation theory and in other triangulated categories. The expansion of all these newly-created tools beyond the known cases is another clear path to be investigated at the meeting.

Among the fascinating developments in the last five years, we could for example mention the progress on the geometric Langlands conjecture. The conjecture asserts the equivalence of two derived categories. Let $G$ be a reductive group with Langlands dual $G'$, and let $X$ be an algebraic curve. The derived categories which should be equivalent are, on the one hand, the derived category of $D$-modules on the moduli stack of $G$-principal bundles on $X$, and, on the other hand, the derived category of quasicoherent sheaves on the $G'$-moduli stack of local systems on $X$. An enormous amount of work has gone into this, by many people (Arikin, BenZvi, Drinfeld, Gaitsgory, Kazhdan,..). and much machinery has been developed, without always interacting with nearby fields. In this conference we plan to correct this: we will have experts in geometric Langlands explain this progress to the others, and also learn from the others about recent progress in triangulated categories that might help them. This should lead to breakthroughs across the board.

In the first edition, we left the door open for un-anticipated new developments which presented an interest beyond their specialized area of discovery and we intend to offer the same flexibility again. This is in line with our overarching objective to let all participants share their particular expertise related to triangulated categories with the entire attendance of the workshop.

Ideally, young researchers will be asked to present the most recent advances, while more experienced colleagues will provide overviews of the uses of triangulated categories in their fields. This formula was well received by the participants of the first edition.

The theory of triangulated categories is relatively simple and elegant. This makes it a most convenient language for formulating results and techniques, and may explain its ubiquity. While of course not everything can be done at the triangular level, what can be accomplished has aesthetic appeal as well as tremendous potential in applications. To illustrate the versatile nature of triangulated categories, let us mention below a few triangular subjects at the center of current research, together with their multidisciplinary applications.

**1. Brown representability theorems.**Originating in stable homotopy, where spectra represent generalized cohomology theories (Brown, Adams, Bousfield), the abstract ``Brown Representability Theorems" (Neeman, Krause) were exported to other subjects, from Grothendieck duality to the theory of motives. These techniques encourage the study of ``large'' triangulated categories, which still needs to be undertaken in several of the areas of expertise that will be represented at the workshop.

**2. Localization, recollements and related techniques.**Verdier localization and Bousfield localization were born in algebraic geometry and topology respectively - they are the main tool for constructing new triangulated categories out of old ones, making them a staple of the theory. The closely related notion of Rickard idempotents is invaluable in modular representation theory of finite groups. Recollements and t-structures were born in geometry (Beilinson, Bernstein, Deligne) and were first applied to the study of perverse sheaves. They are now also extensively used in representation theory and noncommutative algebra. It can also be interesting to classify them (Alonso, Bridgeland, Jeremias, Saorin, Stovicek). The related notion of weight structures, a.k.a. co-t-structures (Bondarko, Pauksztello), has been developed and successfully applied in motivic theory for instance. In that world, the slice filtrations (Voevodsky, Levine) have played an important role. The neighboring formalism of Grothendieck's ``six operations", with its origins in etale cohomology, has now also branched to several fields, including the motivic world (Ayoub).

Putting those ideas at work more systematically throughout mathematics, for instance in representation theory, seems to hold a great potential for new results and interesting connections.

**3. Classification results and support theories.**In several domains, one can find a central result taking the form of the classification of thick subcategories in some triangulated category of interest. Starting in topology with the classification of the thick subcategories of finite spectra (Hopkins, Smith, Devinatz), the idea has been exported to the derived category of a commutative ring (Hopkins, Neeman) and that of a scheme (Thomason), as well as to the stable module category of finite groups (Benson, Carlson, Rickard) and finite group schemes (Friedlander, Pevtsova). In order to distill the underlying ideas, Balmer has developed an axiomatic theory of tensor triangulated categories which allows one to extend algebro-geometric thinking to other fields. Another approach, starting from a commutative ring acting on a large triangulated category (Benson, Iyengar and Krause), has also found applications - notably the extension of the Benson-Carlson-Rickard classification to the large stable module category. A unifying relative version of these ideas (Stevenson) can be applied to new classes of examples such as singularity categories. This is a growing subject with several open questions and very exciting ongoing developments, for instance in equivariant stable homotopy theory. Such results (in progress at present) will be ripe for distribution by the time of the meeting. The equivariant stable homotopy category being in some sense the initial equivariant triangulated category, the results will undoubtedly have repercussions in many fields - from representation theory to motivic stable homotopy theory.

**4. Separable extensions and descent.**A recent generalization of Bousfield localization, analogous to the etale topology generalizing the Zariski topology, is the theory of separable extensions of triangulated categories, as developed by Balmer and his collaborators since the last meeting. For the derived categories of algebraic geometry, separable extensions describe etale extensions of schemes, and recover the etale topology (Neeman). Separable extensions also cover restrictions to subgroups in many equivariant triangulated categories - from KK-theory to equivariant stable homotopy (Dell'Ambrogio, Sanders). These techniques have been combined with descent theory, and in representation theory the combination led to new calculations of endotrivial modules (Balmer, Carlson, Thevenaz). It remains to explore this further, both in representation theory and in other triangulated categories. The expansion of all these newly-created tools beyond the known cases is another clear path to be investigated at the meeting.

**5. The geometric Langlands conjecture.**Among the fascinating developments in the last five years, we could for example mention the progress on the geometric Langlands conjecture. The conjecture asserts the equivalence of two derived categories. Let $G$ be a reductive group with Langlands dual $G'$, and let $X$ be an algebraic curve. The derived categories which should be equivalent are, on the one hand, the derived category of $D$-modules on the moduli stack of $G$-principal bundles on $X$, and, on the other hand, the derived category of quasicoherent sheaves on the $G'$-moduli stack of local systems on $X$. An enormous amount of work has gone into this, by many people (Arikin, BenZvi, Drinfeld, Gaitsgory, Kazhdan,..). and much machinery has been developed, without always interacting with nearby fields. In this conference we plan to correct this: we will have experts in geometric Langlands explain this progress to the others, and also learn from the others about recent progress in triangulated categories that might help them. This should lead to breakthroughs across the board.

**6. The unexpected.**In the first edition, we left the door open for un-anticipated new developments which presented an interest beyond their specialized area of discovery and we intend to offer the same flexibility again. This is in line with our overarching objective to let all participants share their particular expertise related to triangulated categories with the entire attendance of the workshop.