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Hadrian Heine

Publications and source records attributed to Hadrian Heine.

18 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

Stable homotopy theory of higher categories

Stable homotopy theory is governed by the principle that inverting the operation of forming loop spaces produces representing objects for homology theories. We show that this principle is not specific to topology but reflects an intrinsic structural feature of higher category theory: inverting the operation of forming endomorphism $(\infty,\infty)$-categories leads to a stable homotopy theory of higher categories, in which higher categories play the role of spaces and categorical spectra represent homology theories of higher categories. This theory necessarily requires enrichment in the Gray tensor product, reflecting its genuinely higher-categorical nature. Classical stable homotopy theory is recovered by passing to classifying spaces. Its fundamental mechanisms arise as shadows of richer categorical phenomena: stabilization is governed by a categorical Freudenthal suspension theorem, whose classical counterpart arises by passing to classifying spaces. Our main result is a categorical Brown representability theorem classifying homology theories of higher categories by categorical spectra. As a consequence, categorical homology theories give rise to categorical analogues of long exact sequences and to homological algebra of higher categories. As guiding example we study categorical homology, the categorical homology theory whose coeffients are the natural numbers, and prove a Hurewicz theorem and Eilenberg-Zilber theorem.

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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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arXiv2025

Homology of higher categories

The classical Dold--Kan correspondence identifies simplicial abelian groups with connective chain complexes. On the level of homotopy theory, these provide models for strict grouplike $\mathbb{E}_\infty$-spaces and connective $H\mathbb{Z}$-module spectra. We establish a higher-categorical Dold--Kan correspondence identifying stratified simplicial commutative monoids with strict symmetric monoidal $\omega$-categories and with connective categorical $H\mathbb{N}$-module spectra. This provides a combinatorial, algebraic and stable homotopy-theoretic model for higher-categorical homological algebra. As a consequence, the free categorical $H\mathbb{N}$-module spectrum provides a natural notion of homology for higher categories. Through the categorical Dold--Kan correspondence, it is modeled by the free stratified simplicial commutative monoid on the Street nerve, the higher-categorical analogue of singular chains. We obtain a categorical Dold--Thom theorem, which leads to explicit computations of the categorical homology of the globes and shows that categorical homology detects genuinely higher-categorical information invisible to classical homology.

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arXiv2024

A local-global principle for parametrized $\infty$-categories

We prove a local-global principle for $\infty$-categories over any base $\infty$-category $\mathcal{C}$: we show that any $\infty$-category $\mathcal{B} \to \mathcal{C}$ over $\mathcal{C}$ is determined by the following data: the collection of fibers $\mathcal{B}_X$ for $X$ running through the set of equivalence classes of objects of $\mathcal{C}$ endowed with the action of the space of automorphisms $\mathrm{Aut}_X(\mathcal{B})$ on the fiber, the local data, together with a locally cartesian fibration $\mathcal{D} \to \mathcal{C}$ and $\mathrm{Aut}_X(\mathcal{B})$-linear equivalences $\mathcal{D}_X \simeq \mathcal{P}(\mathcal{B}_X)$ to the $\infty$-category of presheaves on $\mathcal{B}_X$, the gluing data. As applications we describe the $\infty$-category of small $\infty$-categories over $[1]$ in terms of the $\infty$-category of left fibrations and prove an end formula for mapping spaces of the internal hom of the $\infty$-category of small $\infty$-categories over $[1]$ and the conditionally existing internal hom of the $\infty$-category of small $\infty$-categories over any small $\infty$-category $\mathcal{C}.$ Considering functoriality in $\mathcal{C}$ we obtain as a corollary that the double $\infty$-category $\mathrm{CORR}$ of correspondences is the pullback of the double $\infty$-category $\mathrm{PR}^L$ of presentable $\infty$-categories along the functor $\infty\mathrm{Cat} \to \mathrm{Pr}^L$ taking presheaves. We deduce that $\infty$-categories over any $\infty$-category $\mathcal{C}$ are classified by normal lax 2-functors.

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arXiv2024

A derived Milnor-Moore theorem

For every stable presentably symmetric monoidal $\infty$-category $\mathcal{C}$ we use the Koszul duality between the spectral Lie operad and the cocommutative cooperad to construct an enveloping Hopf algebra functor $\mathcal{U}: \mathrm{Alg}_{\mathrm{Lie}}(\mathcal{C}) \to \mathrm{Hopf}(\mathcal{C})$ from Lie algebras in $\mathcal{C}$ to cocommutative Hopf algebras in $\mathcal{C}$ left adjoint to a functor of derived primitive elements $\mathrm{Prim}$. We study the unit of this adjunction in rational and chromatic homotopy theory: we prove that if $\mathcal{C}$ is a rational stable presentably symmetric monoidal $\infty$-category, the enveloping Hopf algebra functor $\mathcal{U}: \mathrm{Alg}_{\mathrm{Lie}}(\mathcal{C}) \to \mathrm{Hopf}(\mathcal{C})$ is fully faithful reproving a result of Gaitsgory-Rozenblyum. Let $n \geq 1 $ be a natural and $\Phi[-1]: \mathcal{S}_{v_n} \to \mathrm{Alg}_{\mathrm{Lie}}(\mathrm{Sp}_{T_n})$ the shifted Bousfield-Kuhn functor from $v_n$-periodic homotopy types to spectral Lie algebras in $T_n$-local spectra. We prove that for every $v_n$-periodic homotopy type $X$ the unit $\Phi(X)[-1] \to Prim \mathcal{U}(\Phi(X)[-1])$ identifies with the Goodwillie completion $ \Phi \to \lim_{n \geq 0} P_n(\Phi)$ evaluated at the loop space of $X.$

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arXiv2024

On bi-enriched $\infty$-categories

We extend Lurie's definition of enriched $\infty$-categories to notions of left enriched, right enriched and bienriched $\infty$-categories, which generalize the concepts of closed left tensored, right tensored and bitensored $\infty$-categories and share many desirable features with them. We use bienriched $\infty$-categories to endow the $\infty$-category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched $\infty$-categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we construct enriched Kan-extensions from operadic Kan-extensions, compute the monad for enriched functors, prove an end formula for morphism objects of enriched $\infty$-categories of enriched functors and a coend formula for the relative tensor product of enriched profunctors and construct transfer of enrichment from scalar extension of presentably bitensored $\infty$-categories. In particular, we develop an independent theory of enriched $\infty$-categories for Lurie's model of enriched $\infty$-categories.

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arXiv2024

The higher algebra of weighted colimits

We develop a theory of weighted colimits in the framework of weakly bienriched $\infty$-categories, an extension of Lurie's notion of enriched $\infty$-categories. We prove an existence result for weighted colimits, study weighted colimits of diagrams of enriched functors, express weighted colimits via enriched coends, characterize the enriched $\infty$-category of enriched presheaves as the free cocompletion under weighted colimits, prove a Bousfield-Kan formula for weighted colimits and an enriched adjoint functor theorem and develop a theory of universally adjoining weighted colimits to an enriched $\infty$-category. Via the latter we construct for every presentably $\mathbb{E}_{k+1}$-monoidal $\infty$-category $\mathcal{V}$ for $1 \leq k \leq \infty$ and set $\mathcal{H}$ of weights a presentably $\mathbb{E}_k$-monoidal structure on the $\infty$-category of $\mathcal{V}$-enriched $\infty$-categories that admit $\mathcal{H}$-weighted colimits. Varying $\mathcal{H}$ this $\mathbb{E}_k$-monoidal structure interpolates between the tensor product for $\mathcal{V}$-enriched $\infty$-categories and the relative tensor product for $\infty$-categories presentably left tensored over $\mathcal{V}$. Studying functoriality in $\mathcal{H}$ we deduce that taking $\mathcal{V}$-enriched presheaves is $\mathbb{E}_k$-monoidal with respect to the tensor product on small $\mathcal{V}$-enriched $\infty$-categories and the relative tensor product on $\infty$-categories presentably left tensored over $\mathcal{V}.$ As key applications we construct for every $n \geq 1 $ and set $\mathcal{K}$ of $(\infty, n)$-categories a tensor product for $(\infty,n)$-categories that admit $\mathcal{K}$-indexed (op)lax colimits, a tensor product for Cauchy-complete $\mathcal{V}$-enriched $\infty$-categories and tensor products for (Cauchy complete) $n$-stable, $n$-additive and $n$-preadditive $(\infty,n)$-categories.

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arXiv2023

An equivalence between two models of $\infty$-categories of enriched presheaves

Let $\mathcal{O} \to \mathrm{BM}$ be a $ \mathrm{BM}$-operad that exhibits an $\infty$-category $\mathcal{D}$ as weakly bitensored over non-symmetric $\infty$-operads $\mathcal{V}, \mathcal{W}$ and $\mathcal{C}$ a $\mathcal{V}$-enriched $\infty$-precategory. We construct an equivalence $$\mathrm{Fun}_{\mathrm{Hin}}^{\mathcal{V}}(\mathcal{C},\mathcal{D}) \simeq \mathrm{Fun}^{\mathcal{V}}(\mathcal{C},\mathcal{D}) $$ of $\infty$-categories weakly right tensored over $\mathcal{W}$ between two different models of $\infty$-categories of $\mathcal{V}$-enriched functors, one introduced by Hinich and one constructed by us.

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arXiv2020

An equivalence between enriched $\infty$-categories and $\infty$-categories with weak action

We show that an $\infty$-category $\mathcal{M}$ with a closed left action of a monoidal $\infty$-category $\mathcal{V}$ is completely determined by the $\mathcal{V}$-valued graph of morphism objects equipped with the structure of a $\mathcal{V}$-enrichment in the sense of Gepner-Haugseng. We prove a similar result when $\mathcal{M}$ is a $\mathcal{V}$-enriched $\infty$-category in the sense of Lurie, an operadic generalization of the notion of $\infty$-category with closed left action. Precisely, we prove that sending a $\mathcal{V}$-enriched $\infty$-category in the sense of Lurie to the $\mathcal{V}$-valued graph of morphism objects refines to an equivalence $\chi$ between the $\infty$-category of $\mathcal{V}$-enriched $\infty$-categories in the sense of Lurie and of Gepner-Haugseng. Moreover if $\mathcal{V}$ is a presentably $\mathbb{E}_{\mathrm{k+1}}$-monoidal $\infty$-category for $1 \leq k \leq \infty$, we prove that $\chi$ restricts to a lax $\mathbb{E}_{\mathrm{k}}$-monoidal functor between the $\infty$-category of left $\mathcal{V}$-modules in $\mathrm{Pr}^L$, the symmetric monoidal $\infty$-category of presentable $\infty$-categories, endowed with the relative tensor product, and the tensor product of $\mathcal{V}$-enriched $\infty$-categories of Gepner-Haugseng. As an application of our theory we construct a lax symmetric monoidal embedding of the $\infty$-category of small stable $\infty$-categories into the $\infty$-category of small spectral $\infty$-categories. As a second application we produce a Yoneda-embedding for Lurie's notion of enriched $\infty$-categories.

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arXiv2019

Real K-theory for Waldhausen infinity categories with genuine duality

We develop a new framework to study real $K$-theory in the context of $\infty$-categories. For this, we introduce Waldhausen $\infty$-categories with genuine duality, which will be the input for such $K$-theory. These are Waldhausen $\infty$-categories in the sense of Barwick equipped with a compatible duality and a refinement of their (lax) hermitian objects generalizing the concept of Poincar\'e $\infty$-categories of Lurie. They may also be thought of as a version of complete Segal spaces enriched in genuine $C_2$-spaces whose underlying $\infty$-category carries a compatible Waldhausen structure, since we show that their respective $\infty$-categories are equivalent. We define the real $K$-theory genuine $C_2$-spaces by means of an enriched version of the $S_\bullet$-construction, defined for Waldhausen $\infty$-categories with genuine duality. Moreover, we prove an Additivity Theorem for this $S_\bullet$-construction which leads to an Additivity Theorem for real $K$-theory. Furthermore, such real $K$-theory satisfy a universal property -- analogous to that proved by Barwick for algebraic $K$-theory of Waldhausen $\infty$-categories --: We prove that every theory can be universally turned into an additive theory and identify our real K-theory with the universal additive theory associated to the functor that associates to a Waldhausen $\infty$-category with genuine duality its maximal subspace. Finally, we promote the real $K$-theory genuine $C_2$-spaces to genuine $C_2$-spectra.

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arXiv2017

A duality between monads and monadic morphisms

We establish a duality between monads and monadic morphisms in any $(\infty,2)$-category and characterize monadic morphisms in a wide class of examples. This duality unifies several dualities between algebraic structures and their representations, and provides a general mechanism for transferring structure from a monad to its $\infty$-category of algebras. This transfer of structure yields uniform constructions of tensor products for algebras over lax symmetric monoidal and oplax symmetric monoidal monads, extending classical tensor products for modules and operadic algebras. Using this framework, we construct a relative tensor product for algebras over lax monoidal monads, a tensor product for algebras over Hopf $\infty$-operads and equip the $\infty$-category of operadic algebras with canonical enrichment.

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arXiv2017

A topological model for cellular motivic spectra

For any motivic $\mathbb{E}_\infty$-ring spectrum $A$ we construct an equivalence $\rho$ between the $\infty$-category of cellular motivic $A$-module spectra and modules over an $\mathbb{E}_1$-algebra $\Theta$ in $\mathbb{Z} $-graded spectra, under which the motivic grading corresponds to the $\mathbb{Z}$-grading. If the base is the complex numbers or if $A$ admits an $\mathbb{E}_\infty$-orientation, we refine the $\mathbb{E}_1$-algebra $\Theta$ to an $\mathbb{E}_\infty$-algebra and $\rho$ to a symmetric monoidal equivalence. To capture the symmetric monoidal structure in the general situation, we lift $\rho$ to a symmetric monoidal equivalence to modules over an $\mathbb{E}_\infty$-algebra in $\mathcal{J} $-graded spectra that invert morphisms of $\mathcal{J}$, where $\mathcal{J}$ is the diagram category of Sagave-Schlichtkrull, a model for Quillen's localization of the groupoid of finite sets and bijections.

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arXiv2016

Infinity categories with duality and hermitian multiplicative infinite loop space machines

We show that any preadditive infinity category with duality gives rise to a direct sum hermitian K-theory spectrum. This assignment is lax symmetric monoidal, thereby producing E-infinity ring spectra from preadditive symmetric monoidal infinity categories with duality. To have examples of preadditive symmetric monoidal infinity categories with duality we show that any preadditive symmetric monoidal infinity category, in which every object admits a dual, carries a canonical duality. Moreover we classify and twist the dualities in various ways and apply our definitions for example to finitely generated projective modules over E-infinity ring spectra.

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