arXiv subjects

David Gepner

Publications and source records attributed to David Gepner.

At least 19 recordsLinked to original sources

arXiv2026

Colimits in Oriented Category Theory

In higher category theory, lax colimits are often understood to be a more useful and powerful generalization of usual (homotopy) colimits, which can be recovered from the lax colimit by a suitable localization. However, lax colimits do not provide the correct notion of gluing from the geometric perspective. Indeed, they are incompatible with the notion of categorical dimension, the Gray tensor product, and other basic geometric operations. In this paper, we develop the theory of oriented colimits, which correct the defects of lax colimits, and agree with lax colimits in dimension less than or equal to one. In order to study oriented colimits, we introduce a version of the Grothendieck construction which is compatible with enrichment in the Gray tensor product. We prove that the Grothendieck construction induces an equivalence between cartesian fibrations and presheaves of $(\infty,\infty)$-categories, which is enriched in the Gray tensor product of $(\infty,\infty)$-categories. Oriented colimits simultaneously generalize the concept of lax colimits and the Gray tensor product, and differ from lax colimits in much the same way in which the Gray tensor product differs from the cartesian product. We demonstrate the necessity of oriented colimits by showing that various fundamental constructions in higher category theory fail to be lax colimits but are instances of oriented colimits. As applications, we classify higher-categorical principal bundles, represent higher dimensional adjunctions by bicartesian fibrations of $(\infty,\infty)$-categories, and obtain higher categorical versions of Quillen's Theorems A and B, which admit very natural formulations in our framework.

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arXiv2026

Fibrations in Oriented Category Theory

We study fibrations of higher categories from the perspective of oriented category theory, a framework which accounts for lax phenomena in higher category theory via systematic enrichment in the Gray tensor product. We give several equivalent characterizations of fibrations of $(\infty,\infty)$-categories and oriented categories, and show that categories of fibrations naturally organize to form oriented categories. We study the interaction between fibrations and oriented pullbacks and construct higher-categorical versions of free fibrations and universal fibrations. The latter give rise to Grothendieck constructions for fibrations of $(\infty,\infty)$-categories and oriented categories.

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arXiv2026

An Oriented Street--Roberts Conjecture

We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street--Roberts conjecture, presenting $(\infty,\infty)$-categories as sheaves on suitable families of oriented polytopes, generalizing work of Campion. This allows us to understand $(\infty, \infty)$-categories from a geometric perspective, as directed analogues of homotopy types. These familes of oriented polytopes induce basic operations in higher category theory: for instance, the join, Gray tensor, and bicone arise from the geometry of the orientals, cubes, and orthoplexes, respectively. We study the interaction of these operations and derive some geometric formulae, generalizing work of Ara--Maltsiniotis, Verity, and others.

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arXiv2026

Homotopy Posets, Postnikov Towers, and Hypercompletions of $\infty$-Categories

We show that basic homotopical notions such as homotopy sets and groups, connected and truncated maps, cellular constructions and skeleta, etc., extend to the setting of $(\infty,\infty)$-categories, as well as to presentable categories enriched in $(\infty,\infty)$-categories under the Gray tensor product. The homotopy posets of an $(\infty,\infty)$-category are indexed by boundaries of categorical disks; in particular, there is a fundamental poset for each pair of objects, which we regard as a oriented point where the source and target objects have opposite orientation. In contrast to the situation in topology, weakly contractible geometric building blocks such as oriented polytopes typically have nontrivial homotopy posets. The homotopy posets assemble to form an oriented analogue of the long exact sequence of a fibration and form the layers of a categorical Postnikov tower, which converges for any $(\infty,n)$-category but not for general $(\infty,\infty)$-categories. We show that the full subcategory consisting of the Postnikov complete $(\infty,\infty)$-categories is obtained by inverting the coinductive equivalences and canonically identifies with the limit of the categories of $(\infty,n)$-categories taken along the truncation functors. We also study truncated morphisms in general oriented categories and connected morphisms in presentable oriented categories.

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arXiv2025

Oriented Category Theory

As categorical dimension increases, classical categorical concepts often become too rigid and must be replaced by appropriately lax analogues. A basic manifestation of this principle is that higher-categorical versions of familiar geometric constructions -- such as cylinders, cones, and suspensions -- are not functorial in the classical sense. To remedy this situation, we extend the theory of higher categories to a framework of so-called oriented and antioriented categories, which provides a unified formalism for lax and oplax phenomena and reveals the geometric nature inherent in higher category theory. We characterize (anti)oriented categories as deformations of higher categories in which the various compositions only commute up to coherent (anti)oriented interchange law, and provide a geometric presentation of (anti)oriented categories as sheaves on a suitable thickening of the simplex category. We also construct an embedding of the category of $(\infty,\infty)$-categories into the category of (anti)oriented categories and characterize the image as those $(\infty,\infty)$-categories satisfying a strict interchange law.

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arXiv2024

Global 2-rings and genuine refinements

We introduce the notion of a naive global 2-ring: a functor from the opposite of the $\infty$-category of global spaces to presentably symmetric monoidal stable $\infty$-categories. By passing to global sections, every naive global 2-ring decategorifies to a multiplicative cohomology theory on global spaces, i.e. a naive global ring. We suggest when a naive global 2-ring deserves to be called \emph{genuine}. As evidence, we associate to such a global 2-ring a family of equivariant cohomology theories which satisfy a version of the change of group axioms introduced by Ginzburg, Kapranov and Vasserot. We further show that the decategorified multiplicative global cohomology theory associated to a genuine global $2$-ring canonically refines to an $\mathbb{E}_\infty$-ring object in global spectra. As we show, two interesting examples of genuine global 2-rings are given by quasi-coherent sheaves on the torsion points of an oriented spectral elliptic curve and Lurie's theory of tempered local systems. In particular, we obtain global spectra representing equivariant elliptic cohomology and tempered cohomology.

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arXiv2024

Foams with flat connections and algebraic K-theory

This paper proposes a connection between algebraic K-theory and foam cobordisms, where foams are stratified manifolds with singularities of a prescribed form. We consider $n$-dimensional foams equipped with a flat bundle of finitely-generated projective $R$-modules over each facet of the foam, together with gluing conditions along the subfoam of singular points. In a suitable sense which will become clear, a vertex (or the smallest stratum) of an $n$-dimensional foam replaces an $(n+1)$-simplex with a total ordering of vertices. We show that the first K-theory group of a ring $R$ can be identified with the cobordism group of decorated 1-foams embedded in the plane. A similar relation between the $n$-th algebraic K-theory group of a ring $R$ and the cobordism group of decorated $n$-foams embedded in $\mathbb{R}^{n+1}$ is expected for $n>1$. An analogous correspondence is proposed for arbitrary exact categories. Modifying the embedding and other conditions on the foams may lead to new flavors of K-theory groups.

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arXiv2020

Integral representation theorems for DQ-modules

We identify the type of $\mathbb{C}[[\hbar]]$-linear structure inherent in the $\infty$-categories which arise in the theory of Deformation Quantization modules. Using this structure, we show that the $\infty$-category of quasicoherent cohomologically complete DQ-modules is a deformation of the $\infty$-category of quasicoherent sheaves. We also obtain integral representation results for DQ-modules similar to the ones of To\"en and Ben-Zvi-Nadler-Francis, stating that suitably linear functors between $\infty$-categories of DQ-modules are integral transforms.

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arXiv2020

On equivariant topological modular forms

Following ideas of Lurie, we give in this article a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain in particular equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.

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arXiv2019

The tom Dieck splitting theorem in equivariant motivic homotopy theory

We establish, in the setting of equivariant motivic homotopy theory for a finite group, a version of tom Dieck's splitting theorem for the fixed points of a suspension spectrum. Along the way we establish structural results and constructions for equivariant motivic homotopy theory of independent interest. This includes geometric fixed point functors and the motivic Adams isomorphism.

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arXiv2019

An Introduction to Higher Categorical Algebra

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the derived $\infty$-category of a commutative ring, before turning to our main example, the $\infty$-category of spectra. We then go on to consider ring spectra and their $\infty$-categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and deformation theory.

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arXiv2017

$\infty$-Operads as Analytic Monads

We develop an $\infty$-categorical version of the classical theory of polynomial and analytic functors, initial algebras, and free monads. Using this machinery, we provide a new model for $\infty$-operads, namely $\infty$-operads as analytic monads. We justify this definition by proving that the $\infty$-category of analytic monads is equivalent to that of dendroidal Segal spaces, known to be equivalent to the other existing models for $\infty$-operads.

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arXiv2016

K-theoretic obstructions to bounded t-structures

Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree $-1$. The main results of this paper are that $K_{-1}(E)$ vanishes when $E$ is a small stable $\infty$-category with a bounded t-structure and that $K_{-n}(E)$ vanishes for all $n\geq 1$ when additionally the heart of $E$ is noetherian. It follows that Barwick's theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.

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arXiv2016

Brauer groups and Galois cohomology of commutative ring spectra

In this paper we develop methods for classifying Baker-Richter-Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss-Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$, we find that the "algebraic" Azumaya algebras whose coefficient ring is projective are governed by the Brauer-Wall group of $\pi_0(E)$, recovering a result of Baker-Richter-Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin-Tate spectra have either 4 or 2 Morita equivalence classes depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\Bbb Z/8 \times \Bbb Z/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an "exotic" $KO$-algebra with the same coefficient ring as $End_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$, previously studied by Mathew and Stojanoska.

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arXiv2015

Lax colimits and free fibrations in $\infty$-categories

We define and discuss lax and weighted colimits of diagrams in $\infty$-categories and show that the coCartesian fibration associated to a functor is given by its lax colimit. A key ingredient, of independent interest, is a simple characterization of the free Cartesian fibration associated to a a functor of $\infty$-categories. As an application of these results, we prove that lax representable functors are preserved under exponentiation, and also that the total space of a presentable Cartesian fibration between $\infty$-categories is presentable, generalizing a theorem of Makkai and Par\'e to the $\infty$-categorical setting. Lastly, in the appendix, we observe that pseudofunctors between (2,1)-categories give rise to functors between $\infty$-categories via the Duskin nerve.

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arXiv2014

On localization sequences in the algebraic K-theory of ring spectra

We identify the $K$-theoretic fiber of a localization of ring spectra in terms of the $K$-theory of the endomorphism algebra spectrum of a Koszul-type complex. Using this identification, we provide a negative answer to a question of Rognes for $n>1$ by comparing the traces of the fiber of the map $K(BP(n))\rightarrow K(E(n))$ and of $K(BP(n-1))$ in rational topological Hochschild homology.

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arXiv2014

An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology

We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum $R$, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space $X$ to the infinity category of free rank one $R$-modules, which we show is a model for $BGL_1 R$; we show that $BGL_1 R$ classifies homotopy sheaves of rank one $R$-modules, which we call $R$-line bundles. We use our $R$-module Thom spectrum to define the twisted $R$-homology and cohomology of an $R$-line bundle over a space $X$, classified by a map from $X$ to $BGL_1 R$, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. An earlier version of this paper was part of arXiv:0810.4535.

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arXiv2014

Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory

We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the string orientation of tmf, the spectrum of topological modular forms. We also develop the analogous theory of Thom spectra and orientations for associative ring spectra. Our work is based on a new model of the Thom spectrum as a derived smash product. An earlier version of this paper was part of arXiv:0810.4535.

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