203 citations · 588 across the 21 of their papers we have counts for
23 papers
Perverse schobers of Coxeter type
Tobias Dyckerhoff, Paul Wedrich
We define the concept of an -schober as a categorification of classification data for perverse sheaves on due to Kapranov-Schechtman.…
Lax Additivity
Merlin Christ, Tobias Dyckerhoff, Tashi Walde
We introduce notions of lax semiadditive and lax additive -categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a wel…
N-spherical functors and categorification of Euler's continuants
Tobias Dyckerhoff, Mikhail Kapranov, Vadim Schechtman
Euler's continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their ca…
Complexes of stable -categories
Merlin Christ, Tobias Dyckerhoff, Tashi Walde
We study complexes of stable -categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including repre…
Spherical adjunctions of stable -categories and the relative S-construction
Tobias Dyckerhoff, Mikhail Kapranov, Vadim Schechtman +1
We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable -categories. Building on this, we study the relative Waldhausen S-…
Perverse sheaves on Riemann surfaces as Milnor sheaves
Tobias Dyckerhoff, Mikhail Kapranov, Yan Soibelman
Constructible sheaves of abelian groups on a stratified space can be equivalently described in terms of representations of the exit-path category. In this work, we provide a simila…
The symplectic geometry of higher Auslander algebras: Symmetric products of disks
Tobias Dyckerhoff, Gustavo Jasso, Yanki Lekili
We show that the perfect derived categories of Iyama's -dimensional Auslander algebras of type are equivalent to the partially wrapped Fukaya categories of the -…
A relative 2-nerve
Fernando Abellán García, Tobias Dyckerhoff, Walker H. Stern
In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to -categoric…
Generalised BGP reflection functors via the Grothendieck construction
Tobias Dyckerhoff, Gustavo Jasso, Tashi Walde
Inspired by work of Ladkani, we explain how to construct generalisations of the classical reflection functors of Bernšte\uın, Gel'fand and Ponomarev by means of the Grothendieck co…
Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects
Christopher Brav, Tobias Dyckerhoff
We show that a Calabi-Yau structure of dimension on a smooth dg category induces a symplectic form of degree on the moduli space of objects . We show moreover…
Simplicial structures in higher Auslander-Reiten theory
Tobias Dyckerhoff, Gustavo Jasso, Tashi Walde
We develop a novel combinatorial perspective on the higher Auslander algebras of type , a family of algebras arising in the context of Iyama's higher Auslander-Reiten t…
A categorified Dold-Kan correspondence
Tobias Dyckerhoff
In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined -categories of simplicial stab…
Topological K-theory of Equivariant Singularity Categories
Michael K. Brown, Tobias Dyckerhoff
We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg…
Relative Calabi-Yau structures
Christopher Brav, Tobias Dyckerhoff
We introduce relative noncommutative Calabi-Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geo…
Higher categorical aspects of Hall Algebras
Tobias Dyckerhoff
These are extended notes for a series of lectures on Hall algebras given at the CRM Barcelona in February 2015. The basic idea of the theory of Hall algebras is that the collection…
A1-homotopy invariants of topological Fukaya categories of surfaces
Tobias Dyckerhoff
We provide an explicit formula for localizing -homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential $…
Crossed simplicial groups and structured surfaces
Tobias Dyckerhoff, Mikhail Kapranov
We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the…
Triangulated surfaces in triangulated categories
Tobias Dyckerhoff, Mikhail Kapranov
For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangl…
Higher Segal spaces I
Tobias Dyckerhoff, Mikhail Kapranov
This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplici…
Pushing forward matrix factorisations
Tobias Dyckerhoff, Daniel Murfet
We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the…
The Kapustin-Li formula revisited
Tobias Dyckerhoff, Daniel Murfet
We provide a new perspective on the Kapustin-Li formula for the duality pairing on the morphism complexes in the matrix factorization category of an isolated hypersurface singulari…
Compact generators in categories of matrix factorizations
Tobias Dyckerhoff
We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by…
The inverse problem of differential Galois theory over the field R(z)
Tobias Dyckerhoff
We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extens…