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Tobias Dyckerhoff

Publications and source records attributed to Tobias Dyckerhoff.

At least 19 recordsLinked to original sources
arXiv2025

Lax Additivity

We introduce notions of lax semiadditive and lax additive $(\infty,2)$-categories, categorifying the classical notions of semiadditive and additive 1-categories. To establish a well-behaved axiomatic framework, we develop a calculus of lax matrices and use it to prove that in locally cocomplete $(\infty,2)$-categories lax limits and lax colimits agree and are absolute. In the lax additive setting, we categorify fundamental constructions from homological algebra such as mapping complexes and mapping cones and establish their basic properties.

math.CT↗
arXiv2025

Hypersheaves and bases

Let $X$ be a topological space equipped with a basis. We prove that, for every $\infty$-category $\mathcal{C}$ with limits, the restriction functor from $\mathcal{C}$-valued hypersheaves on $X$ to basic hypersheaves is an equivalence of $\infty$-categories.

math.CT↗
arXiv2025

Cyclic polytopes, orientals, and correspondences: some aspects of higher Segal spaces

We discuss the role of higher Segal spaces at the interface of cyclic polytopes, orientals, and higher correspondences. Along the way we review examples from algebraic K-theory, show how cyclic polytopes provide a geometric model for the definition of orientals, and establish a characterization of higher Segal spaces as lax monadic structures in higher correspondence categories.

math.AT↗
arXiv2025

Perverse schobers of Coxeter type $\mathbb{A}$

We define the concept of an $\mathbb{A}_n$-schober as a categorification of classification data for perverse sheaves on $\mathrm{Sym}^{n+1}(\mathbb{C})$ due to Kapranov-Schechtman. We show that any $\mathbb{A}_n$-schober gives rise to a categorical action of the Artin braid group $\mathrm{Br}_{n+1}$ and demonstrate how this recovers familiar examples of such actions arising from Seidel-Thomas $\mathbb{A}_n$-configurations of spherical objects in categorical Picard-Lefschetz theory and Rickard complexes in link homology theory. As a key example, we use singular Soergel bimodules to construct a factorizing family of $\mathbb{A}_n$-schobers which we refer to as Soergel schobers. We expect such families to give rise to a categorical analog of a graded bialgebra valued in a suitably defined freely generated braided monoidal $(\infty,2)$-category.

math.QA↗
arXiv2024

Complexes of stable $\infty$-categories

We study complexes of stable $\infty$-categories, referred to as categorical complexes. As we demonstrate, examples of such complexes arise in a variety of subjects including representation theory, algebraic geometry, symplectic geometry, and differential topology. One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions. As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context. We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data. A variant of homological mirror symmetry for categorical complexes is proposed and verified for $\mathbb{C}\mathrm{P}^2$.

math.AG↗
arXiv2023

N-spherical functors and categorification of Euler's continuants

Euler's continuants are universal polynomials expressing the numerator and denominator of a finite continued fraction whose entries are independent variables. We introduce their categorical lifts which are natural complexes (more precisely, coherently commutative cubes) of functors involving compositions of a given functor and its adjoints of various orders, with the differentials built out of units and counits of the adjunctions. In the stable infinity-categorical context these complexes/cubes can be assigned totalizations which are new functors serving as higher analogs of the spherical twist and cotwist. We define N-spherical functors by vanishing of the twist and cotwist of order N-1 in which case those of order N-2 are equivalences. The usual concept of a spherical functor corresponds to N=4. We characterize N-periodic semi-orthogonal decompositions of triangulated (stable infinity-) categories in terms of N-sphericity of their gluing functors. The procedure of forming iterated orthogonals turns out to be analogous to the procedure of forming a continued fraction.

math.CT↗
arXiv2021

Spherical adjunctions of stable $\infty$-categories and the relative S-construction

We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and equip it with a natural paracyclic structure (``rotational symmetry''). This fulfills a part of the general program to provide a rigorous account of perverse schobers which are (thus far conjectural) categorifications of perverse sheaves. Namely, in terms of our previous identification of perverse sheaves on Riemann surfaces with Milnor sheaves, the relative $S$-construction with its paracyclic symmetry amounts to a categorification of the stalks of a Milnor sheaf at a singularity of the corresponding perverse sheaf. The action of the paracyclic rotation is a categorical analog of the monodromy on the vanishing cycles of a perverse sheaf. Having this local categorification in mind, we may view the S-construction of a spherical functor as defining a schober locally at a singularity. Each component $S_n(F)$ can be interpreted as a partially wrapped Fukaya category of the disk with coefficients in the schober and with $n+1$ stops at the boundary.

math.AG↗
arXiv2021

Perverse sheaves on Riemann surfaces as Milnor sheaves

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 similar presentation of the abelian category of perverse sheaves on a stratified surface in terms of representations of the so-called paracyclic category of the surface. The category models a hybrid exit-entrance behaviour with respect to chosen sectors of direction, placing it "in between" exit and entrance path categories. In particular, this perspective yields an intrinsic definition of perverse sheaves as an abelian category without reference to derived categories and t-structures.

math.AG↗
arXiv2020

A relative 2-nerve

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 $\infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $\infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.

math.AT↗
arXiv2019

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

We show that the perfect derived categories of Iyama's $d$-dimensional Auslander algebras of type $\mathbb{A}$ are equivalent to the partially wrapped Fukaya categories of the $d$-fold symmetric product of the $2$-dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the $d$-fold symmetric product of the disk and those of its $(n-d)$-fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type $\mathbb{A}$. As a byproduct of our results, we deduce that the partially wrapped Fukaya categories associated to the $d$-fold symmetric product of the disk organise into a paracyclic object equivalent to the $d$-dimensional Waldhausen $\operatorname{S}$-construction, a simplicial space whose geometric realisation provides the $d$-fold delooping of the connective algebraic $K$-theory space of the ring of coefficients.

math.SG↗
arXiv2019

Topological K-theory of Equivariant Singularity Categories

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 singularity category of a weighted projective hypersurface in terms of its Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah-Bott-Shapiro construction to the level of spectra.

math.AG↗
arXiv2018

Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects

We show that a Calabi-Yau structure of dimension $d$ on a smooth dg category $C$ induces a symplectic form of degree $2-d$ on the moduli space of objects $M_{C}$. We show moreover that a relative Calabi-Yau structure on a dg functor $C \to D$ compatible with the absolute Calabi-Yau structure on $C$ induces a Lagrangian structure on the corresponding map of moduli $M_{D} \to M_{C}$.

math.AG↗
arXiv2018

Simplicial structures in higher Auslander-Reiten theory

We develop a novel combinatorial perspective on the higher Auslander algebras of type $\mathbb{A}$, a family of algebras arising in the context of Iyama's higher Auslander-Reiten theory. This approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to Eilenberg-MacLane spaces and higher-dimensional versions of Waldhausen's $\operatorname{S}_\bullet$-construction in algebraic $K$-theory. As an application of our techniques we provide a generalisation of the higher reflection functors of Iyama and Oppermann to representations with values in stable $\infty$-categories. The resulting combinatorial framework of slice mutation can be regarded as a higher-dimensional variant of the abstract representation theory of type $\mathbb{A}$ quivers developed by Groth and Šťov\'ıček. Our simplicial point of view then naturally leads to an interplay between slice mutation, horn filling conditions, and the higher Segal conditions of Dyckerhoff and Kapranov. In this context, we provide a classification of higher Segal objects with values in any abelian category or stable $\infty$-category.

math.RT↗
arXiv2017

A categorified Dold-Kan correspondence

In this work, we establish a categorification of the classical Dold-Kan correspondence in the form of an equivalence between suitably defined $\infty$-categories of simplicial stable $\infty$-categories and connective chain complexes of stable $\infty$-categories. The result may be regarded as a contribution to the foundations of an emerging subject that could be termed categorified homological algebra.

math.AT↗
arXiv2016

Relative Calabi-Yau structures

We introduce relative noncommutative Calabi-Yau structures defined on functors of differential graded categories. Examples arise in various contexts such as topology, algebraic geometry, and representation theory. Our main result is a composition law for Calabi-Yau cospans generalizing the classical composition of cobordisms of oriented manifolds. As an application, we construct Calabi-Yau structures on topological Fukaya categories of framed punctured Riemann surfaces.

math.AG↗
arXiv2016

A1-homotopy invariants of topological Fukaya categories of surfaces

We provide an explicit formula for localizing $A^1$-homotopy invariants of topological Fukaya categories of marked surfaces. Following a proposal of Kontsevich, this differential $\mathbb Z$-graded category is defined as global sections of a constructible cosheaf of dg categories on any spine of the surface. Our theorem utilizes this sheaf-theoretic description to reduce the calculation of invariants to the local case when the surface is a boundary-marked disk. At the heart of the proof lies a theory of localization for topological Fukaya categories which is a combinatorial analog of Thomason-Trobaugh's theory of localization in the context of algebraic K-theory for schemes.

math.CT↗
arXiv2015

Crossed simplicial groups and structured surfaces

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 necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.

math.AT↗