Inspired by the work of P. Bubenik et al., we define exact weights on objects in a triangulated category. In particular, we extend the concept of path metrics for abelian categories to triangulated categories and discuss some of their properties. We are particularly interested in triangulated categories induced by Frobenius categories.
I will talk about two notions of dimension of a triangulated category. The first one is the classical Rouquier dimension, based on generation time with respect to a generator, while the second one is the more recent concept of Serre dimension, based on behavior of iterations of the Serre functor. I will propose 'ideal' properties of dimension that one would like to have, and compare them to properties of Rouquier and Serre dimension, both known and conjectural. Various examples of categories where dimension is known will be given and discussed.
Finite tensor categories are important generalizations of the categories of finite-dimensional modules of finite-dimensional Hopf algebras. There are two support theories for them, the cohomological one and one based on the noncommutative Balmer spectrum of the corresponding stable module category. We will describe general results linking the two types of support via a new notion of categorical center of the cohomology ring of a finite tensor category and will state a conjecture giving the exact relation. The construction and results will be illustrated with various examples.
A quasi-abelian category is an additive category with all kernels and cokernels, along with some additional conditions allowing us to extend notions from homological algebra to them. A key example is the category of complete bornological spaces which is derived equivalent to the category of inductive limits of Banach spaces. In this talk, we will introduce the key concepts in the theory of quasi-abelian categories and we will discuss their potential applications. In particular, we will see how we can extend ideas from Koszul duality to quasi-abelian categories, as well as their use more generally as a setting for a new theory of derived analytic geometry proposed by my supervisor Kobi Kremnizer and his collaborators.
Let T be a suitable triangulated category and C a full subcategory of T closed under summands and extensions. An indecomposable object c in C is called Ext-projective if Ext1(c,C)=0. Such an object cannot appear as the endterm of an Auslander-Reiten triangle in C. However, if there exists a minimal right almost split morphism b→c in C, then the triangle x→b→c→ extending it is a so called left-weak Auslander-Reiten triangle in C. We show how in some cases removing the indecomposable c from the subcategory C and replacing it with the indecomposable x gives a new extension closed subcategory C' of T and see how this operation is related to Iyama-Yoshino mutation of C with respect to a rigid subcategory. Time permitting, we will see the application of the result to cluster categories of type A.
Various notions of support have been studied in representation theory (by Carlson, Snashall-Solberg, Balmer, Benson-Iyengar-Krause, Friedlander-Pevtsova, Nakano-Vashaw-Yakimov, to name only few). My talk offers some new and unifying perspective: For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way we establish Mayer-Vietoris sequences for pairs of central subcategories.
The derived category, D(A), of the category Mod(A) of modules over a ring A is an important example of a triangulated category in algebra. It can be obtained as the homotopy category of the category Ch(A) of chain complexes of A-modules equipped with its standard model structure. One can view Ch(A) as the category Fun(Q, Mod(A)) of additive functors from a certain small preadditive category Q to Mod(A). The model structure on Ch(A) = Fun(Q, Mod(A)) is not inherited from a model structure on Mod(A) but arises instead from the "self-injectivity" of the special category Q. We will show that the functor category Fun(Q, Mod(A)) has two interesting model structures for many other self-injective small preadditive categories Q. These two model structures have the same weak equivalences, and the associated homotopy category is what we call the Q-shaped derived category of A. We will also show that it is possible to generalize the homology functors on Ch(A) to homology functors on Fun(Q, Mod(A)) for most self-injective small preadditive categories Q.
In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with a cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained with Idun Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on recent progress in the general case obtained in joint work with Junyang Liu and based on Van den Bergh's structure theorem for complete Calabi-Yau algebras.
After reviewing the basic definitions of tensor categories and the notion of semisimplification of symmetric tensor categories, it will be shown how the semisimplification of the category of representations of the cyclic group of order 3 over a field of characteristic 3 is naturally equivalent to the category of vector superspaces over this field. This allows to define a superalgebra starting with any algebra endowed with an order 3 automorphism. As a noteworthy example, the exceptional composition superalgebras will be obtained, in a systematic way, from the split octonion algebra.
The operation of mutation has a long history in representation theory and algebraic geometry, be it in the context of exceptional collections of sheaves or in the combinatorial study of tilting modules. The aim is to create a new object from an old one by changing a designated part of it and keeping the other part. Here I discuss a variant in the context of cosilting objects in compactly generating triangulated categories (which are also known as derived injective cogenerators for t-structures of Grothendieck type). In that case, the operation of mutation corresponds to certain nice tilts of t-structures with respect to torsion pairs. Time permitting, I will explain how this is related to the lattice of torsion pairs in the category of finite-dimensional modules over a finite-dimensional algebra, as studied by Demonet, Iyama, Reading, Reiten and Thomas.
Motivated by the work of Cohn and Schofield on Sylvester rank functions on rings, Chuang and Lazarev have recently introduced the notion of a rank function on a triangulated category. It turns out that a rank function on a category C can be recast as translation-invariant additive function on its abelianisation mod C. As a consequence, integral rank functions have a unique decomposition into irreducible ones, and they are related to a number of important concepts associated to the localisation theory of mod C. When C is the subcategory of compact objects of a compactly generated triangulated category T, these connections become particularly nice and provide a link between rank functions on C and smashing localisations of T. In particular, when C is the perfect derived category per(A) of a DG algebra A, this allows us to classify homological epimorphisms from A to B with per(B) locally finite via special rank functions, extending a result of Chuang and Lazarev.
Enriques categories are characterized by the property that their Serre functor is the composition of an involutive autoequivalence and the shift by 2. The bounded derived category of an Enriques surface is an example of Enriques category. Other interesting examples are provided by the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids.
In this talk, we study moduli spaces of semistable objects in the Kuznetsov components of Gushel-Mukai threefolds and quartic double solids with respect to Serre-invariant stability conditions. We provide a result of non-emptiness for these moduli spaces, by using the relation with certain moduli spaces on the associated K3 category.
I will speak about adjacent weight and t-structures (this means that either left or 'right hand side halves' of these 'structures' w and t coincide) in triangulated categories. In particular, for any compactly generated t there exists a weight structure w right adjacent to it. This yields injective cogenerators for the heart of t; it follows that the heart of t is Grothendieck abelian. To construct this w I proved that any perfect set of objects (in a smashing triangulated category) generates a weight structure w. Moreover, if a triangulated category satisfies the Brown representability property then t that is left adjacent to w exists if and only if w is smashing (i.e., coproducts respect weight decompositions).
In the category of modules over a ring, purity may be viewed as a weakening of splitting - a short exact sequence is pure if and only if it is split exact after applying the character dual. The notion of purity in triangulated categories was introduced by Krause, and it has since been seen to be intimately related to many questions of interest in representation theory and homotopy theory. However, in general, it can be hard to check whether a class is closed under purity operations. In this talk, I will explain a framework of duality pairs in triangulated categories which provides an elementary way to check pure closure properties, and illustrate this with a range of examples, often from the tensor-triangular perspective. I will also discuss an application to the study of definable subcategories of triangulated categories.
When studying the Adams spectral sequence in triangulated categories, one runs into the issue of choosing suitably coherent cofibers in an Adams resolution. Motivated by this, in joint work with Dan Christensen, we develop tools to deal with the limited coherence afforded by the triangulated structure. We use and expand Neeman's work on good morphisms of exact triangles. The talk will include examples from stable module categories of group algebras.
Inspired by the intrinsic formality of graded algebras, we give a characterization of strongly unique DG-enhancements for a large class of algebraic triangulated categories, linear over a commutative ring. We will discuss applications to bounded derived categories and bounded homotopy categories of complexes. For the sake of an example, the bounded derived category of finitely generated abelian groups has a strongly unique enhancement.
