The heart fan is a new convex-geometric invariant of an abelian category which captures interesting aspects of the related homological algebra. I will review the construction and some of its key properties, illustrating them through examples. In particular, I will explain how the heart fan can be viewed as a 'universal phase diagram' for Bridgeland stability conditions with the given heart.
Many results about schemes can be generalized to the non-commutative setting of stable ∞-categories. For bounded weighted categories, things are even better: one can formulate and prove a natural analogue of the theorem of Dundas-Goodwillie-McCarthy which is one of the fundamental tools in studying algebraic K-theory. However, in algebraic geometry bounded weighted categories do not show up very often: for instance, the ∞-category of perfect complexes over a scheme X only admits a reasonable weight structure when X is affine. We introduce a new notion of a c-category which is designed to cover a diverse class of geometric examples, including all quasi-compact quasi-separated schemes, yet allowing for all the weighted arguments to work out in this setting. Our main result shows that a c-category can be resolved in finitely many steps by categories of perfect complexes over connective ring spectra. This allows us to prove an analogue of the DGM theorem for c-categories, as well as the vanishing of their Hochschild homology below a certain degree. We also show that either of the following admits a structure of a c-category:
1. the derived category of any exact (∞-)category that has finite Ext-dimension;
2. the subcategory of compact objects in any weakly approximable stable ∞-category in the sense of Neeman.
In particular, using the computation of the Hochschild homology for c-categories mentioned before we obtain that the category of module-spectra over the ring C*(𝕊2) of cochains over the 2-sphere is not weakly approximable.
Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.
Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.
One way to study triangulated categories is through finite building. An object X finitely builds an object Y, if Y can be obtained from X by taking cones, suspensions and retracts. The X-level measures the number of cones required in this process; this can be thought of as the generation time. I will explain the behaviour of level with respect to tensor products and other biexact functors for enhanced triangulated categories. I will further present applications to the level of Koszul objects.
The purpose of this talk is to present an ongoing work on geometric quantization in the setting of shifted symplectic structures. I will start by recalling the various notions involved as well as the results previously obtained by James Wallbridge, who constructed the prequantized (higher) categories of a given integral shifted symplectic structure. I will then explain our main result so far: the construction of the shifted analogues of the Kostant–Souriau prequantum operators, which will be realized as a "Poisson module over a Poisson category" (a categorification of the notion of a Poisson module over a Poisson algebra). This will be obtained by means of deformation theory arguments for categories of sheaves in the setting of (derived) differential geometry. If time permits, I will discuss further aspects associated to the notion of polarizations of shifted symplectic structures.
Carlson's connectedness theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.
Lawvere’s Elementary Theory of the Category of Sets (ETCS) posits that the category Set is a well-pointed elementary topos with natural numbers object satisfying the axiom of choice. This provides a category theoretic foundation for mathematics which axiomatises the properties of function composition in contrast to Zermelo-Fraenkel set theory with the axiom of choice (ZFC), which axiomatises sets and their membership relation. Furthermore, ETCS augmented with the axiom schema of replacement can be shown to be equiconsistent with ZFC.
In this talk, I will present a categorification of ETCS which axiomatises the 2-category of small categories, functors and natural transformations; this is the elementary theory of the 2-category of small categories (ET2CSC) of the title. This extends Bourke’s characterisation of categories internal to a category E with pullbacks to the setting where E satisfies the extra properties of ETCS. Important 2-categorical definitions I will introduce are 2-well-pointedness, the full subobject classifier and the categorified axiom of choice. The main conclusion is that ET2CSC is 'Morita biequivalent’ with ETCS, meaning that the two theories have biequivalent 2-categories of models.
I will also describe how Shulman and Weber’s ideas on discrete opfibration classifiers can be used to incorporate replacement, in a way reminiscent of algebraic set theory.
The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.
I’ll define what it means for a tensor triangulated category to be locally regular, and discuss this condition and its implications for the stable category of a finite group.
Carlson’s Connectedness Theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. For monoidal triangulated categories, the Balmer spectrum is an intrinsic geometric space associated to the category which generalizes the notion of cohomological support for finite groups. In this talk, we will discuss a generalization of the Carlson Connectedness Theorem: that the Balmer support of any indecomposable object in a monoidal triangulated category with a thick generator is a connected subset of the Balmer spectrum. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.
Describing the thick ideals of a monoidal triangulated category is a key component of the analysis of the category. We will show how this can be done by non-commutative tensor triangular geometry (NTTG), thus extending the celebrated Balmer’s theorem from the symmetric case. We will then use NTTG to analyse the stable categories of finite tensor categories, which play an important role in representation theory, mathematical physics and quantum computing. We will present general results linking this approach to the traditional one through cohomological support, based on a notion of categorical centers of cohomology rings of monoidal triangulated categories.
In the 90s, Nakajima and Grojnowski identified the total cohomology of the Hilbert schemes of points on a smooth projective surface with the Fock space representation of the Heisenberg algebra associated to its cohomology lattice. Later, Krug lifted this to derived categories and generalized it to the symmetric quotient stacks of any smooth projective variety.
On the other hand, Khovanov introduced a categorification of the free boson Heisenberg algebra, i.e., the one associated to the rank 1 lattice. It is a monoidal category whose morphisms are described by a certain planar diagram calculus which categorifies the Heisenberg relations. A similar categorification was constructed by Cautis and Licata for the Heisenberg algebras of ADE type root lattices.
We show how to associate the Heisenberg 2-category to any smooth and proper DG category and then define its Fock space 2-representation. This construction unifies all the results above and extends them to what can be viewed as the generality of arbitrary non-commutative smooth and proper schemes.
In this talk, we show that the homotopy category of (small) dg categories and the homotopy category of A∞-categories are equivalent (even from a higher categorical viewpoint). We will discuss several issues related to the various notions of unity and provide several applications. The main ones are about the uniqueness of enhancements for triangulated categories and a full proof of a claim by Kontsevich and Keller concerning a description of the category of internal Homs for dg categories.
Pollitz gave a characterization of complete intersection rings in terms of the triangulated structure of their derived category, akin to the Auslander-Buchsbaum-Serre characterization of regular rings. In this talk, we will explore how to bring this characterization back to the world of modules, and discuss the role of cohomological support varieties in solving this problem.
The goal of this talk will be to present the results from my recent joint work with Vova Sosnilo and Christoph Winges, where we prove that every spectrum is the (non-connective) K-theory spectrum of a stable category. Our main application of this is the disproof of a conjecture by Antieau-Gepner-Heller about a non-connective version of the theorem of the heart in the non-noetherian setting; but I will also try to mention other perspectives on this result.
We discuss reflectivity of colocalizing subcategories of triangulated categories under suitable set-theoretical assumptions. In earlier joint work with Gutierrez and Rosicky, we proved that if K is any locally presentable category with a stable model category structure, then Vopenka's principle implies that every full subcategory L of the homotopy category of K closed under products and fibres is reflective. Moreover, if L is colocalizing, then the reflection is exact. Using recent progress in large-cardinal theory, we show that the statement that every full subcategory closed under products and fibres is reflective is, in fact, equivalent to the so-called weak Vopenka principle. Hence this statement cannot be proved using only the ZFC axioms.
We consider the (infinity) category of excisive (aka polynomial) functors from Spectra to Spectra. Understanding this category is a basic problem in functor calculus. We will approach it from the perspective of tensor triangular geometry. Day convolution equips the category of excisive functors with the structure of a rigid monoidal triangulated category. We describe completely the Balmer spectrum of this category, i.e., its spectrum of prime tensor ideals. This leads to a Thick Subcategory Theorem for excisive functors. A key ingredient in the proof is a blueshift theorem for the generalized Tate construction associated with the family of non-transitive subgroups of products of symmetric groups. If there is time, I will say something about work in progress to extend these results to more general functor categories.
Conilpotent Koszul duality, as formulated by Positselski and Lefevre-Hasegawa, gives an equivalence (of model categories, or of ∞-categories) between augmented dg algebras and conilpotent dg-coalgebras. One should think of this as a non-commutative version of the Lurie-Pridham correspondence: indeed in characteristic zero, cocommutative conilpotent dg coalgebras are Koszul dual to dg Lie algebras, and this is precisely the correspondence between formal moduli problems and their tangent complexes. I'll talk about a global analogue where the conilpotency assumption is removed; geometrically this corresponds to non-commutative formal moduli problems modelled on profinite completions, rather than pro-Artinian completions. Global Koszul duality is best expressed as a Quillen equivalence between curved dg algebras and curved dg coalgebras, and in both categories the weak equivalences are defined using an auxiliary object, the Maurer-Cartan dg category of a curved dg algebra.
