# TTT118 Sheffield

## Date Thursday 1st February 2024

## Speakers

The speakers will be Rudradip Biswas (Warwick), Elena Caviglia (Leicester), Tom Peirce (Warwick) and Callum Reader (Sheffield).

Programme

All talks will be in J11, floor J of the Hicks Building, Hounsfield Road, University of Sheffield S3 7RH.

The tea/coffee breaks will be in the common room on I floor of the Hicks Building.

11:30-12:00 tea/coffee

12:00-13:00 Callum Reader (Sheffield)

13:00-14:15 lunch

14:15-14:45 Elena Caviglia (Leicester)

14:45-15:15 Tom Peirce (Warwick) [slides]

15:15-16:00 tea/coffee

16:00-17:00 Rudradip Biswas (Warwick)

Titles and abstracts

Rudradip Biswas

Bounded t-structures and singularity categories of triangulated categories.

Abstract

Recently, Amnon Neeman settled a bold conjecture by Antieau, Gepner, and Heller regarding the relationship between the regularity of finite-dimensional noetherian schemes and the existence of bounded t-structures on their derived categories of perfect complexes [2].

In a new paper [1], with different methods, we prove some very general results about the existence of bounded t-structures on general triangulated categories and their invariance under "completion". Our treatment, when specialized to the case of schemes, immediately gives us Neeman's theorem as an application and significantly generalizes another remarkable theorem by Neeman about the equivalence of bounded t-structures on the bounded derived categories of coherent sheaves. When specialized to other cases like (not necessarily commutative) rings, nonpositive DG-rings, connective E_1-rings, triangulated categories without models, etc., we get many other applications and examples.

References:

[1] R. Biswas, H. Chen, K. Manali Rahul, C. Parker, and J. Zheng. "Bounded t-structures, finitistic dimensions, and singularity categories of triangulated categories." arXiv:2401.00130

[2] A. Neeman. "Bounded t-structures on the category of perfect complexes." Acta Math, to appear.

Elena Caviglia

Generalizing principal bundles and quotient stacks

Abstract

Principal bundles over topological spaces are an important and useful notion in geometry and topology. We will capture this notion in a categorical way and produce a new concept of principal bundle that makes sense in any nice category equipped with a Grothendieck topology. The topological group involved in the standard notion of principal bundle is generalized with a group object in the category. And locally trivial morphisms are generalized considering pullbacks along the morphisms of a covering family for the Grothendieck topology.

We will then see how to use generalized principal bundles to construct generalized quotient prestacks. When the Grothendieck topology is subcanonical and the category is nice enough, generalized quotient prestacks are stacks.

We will then move to dimension 2, introducing a notion of principal 2-bundle that makes sense in any nice 2-category equipped with a Grothendieck bitopology. In this context the group object involved in the definition is generalized with an internal 2-group and pullbacks are replaced with comma objects.

Finally, we will use principal 2-bundles to construct a generalization of quotient prestacks one dimension higher. When the Grothendieck bitopology is subcanonical, these objects are instances of an original notion of 2-stack.

Tom Peirce

The Rational Nucleus of a Compact Lie Group

Abstract

The nucleus of a finite group G, defined by Benson-Carlson-Robinson, is an object in modular representation theory linking structural properties of the group to the vanishing of cohomology of certain representations. This talk describes an adaption of its definition to compact Lie groups in characteristic zero. We show that it can be described purely in terms of the action of the Weyl group on a maximal torus, and hence also by the commutative algebra of the cohomology ring H*(BG). We then relate this object to the singularity category of the differential-graded algebra C*(BG), proving an analogous result to a recent conjecture by Benson-Greenlees for finite groups.

Callum Reader

(Co)Traces and Enrichment in Bicategories

Abstract

More than half a century ago, Kelly introduced the concept of a compact closed category, a type of symmetric monoidal category where we can define the `trace' of a morphism. For the category of finite dimensional vector spaces this recovers the usual notion of trace, but unfortunately there are very few other interesting examples.

If we broaden our horizons, and instead consider compact closed bicategories, then we find many interesting examples: sets and relations, algebras and bimodules, differential graded algebras and bimodules, categories and profunctors to name just a few. The traces in these particular examples give us a reflexivity indicator, coinvariants, Hochschild homology, and ends respectively.

In this talk we discuss how to define the dual notion of a cotrace in a bicategory. We show that, in very many examples, the cotrace recovers something dual to the trace. We also show that the existence of a cotrace allows us to enrich our bicategory in a way that replaces 2-cells with `scalars', and that this enrichment behaves analogously to the Frobenius inner product for matrices. Finally, we give some general properties of the trace and cotrace that help explain the behaviour of Hochschild (co)homology, and suggest that this framework could be used to study other (co)homology theories, such as THH.

## Further information

Participants from the UK nodes can claim travel expenses. There are links to the form on the main TTT page and paper copies will be available on the day.

We will go for drinks and an early dinner in the evening.