arXiv subjects

Peter J. Haine

Publications and source records attributed to Peter J. Haine.

At least 19 recordsLinked to original sources

arXiv2026

The derived moduli of perverse sheaves

We construct higher derived Artin stacks parametrizing constructible sheaves on complex algebraic varieties and compact real analytic varieties. Furthermore, we show that every perversity function gives rise to an open substack of perverse sheaves, which is a 1-Artin stack locally of finite presentation that generalizes usual character stacks. As a sample application of the derived structure, we construct new examples of cohomological Hall algebras associated to punctured Riemann surfaces.

math.AG↗

arXiv2026

Exodromy beyond conicality

We show that compact subanalytic stratified spaces and algebraic stratifications of real varieties have finite exit-path $\infty$-categories, refining classical theorems of Lefschetz-Whitehead, Lojasiewicz, and Hironaka on the finiteness of the underlying homotopy types of these spaces. These stratifications are typically not conical; hence we cannot rely on the currently available exodromy equivalence between constructible sheaves on a stratified space, which requires conicality as a fundamental hypothesis. Building on ideas of Clausen and Jansen, we study the class of exodromic stratified spaces, for which the conclusion of the exodromy theorem holds. We prove two new fundamental properties of this class of stratified spaces: coarsenings of exodromic stratifications are exodromic, and every morphism between exodromic stratified spaces induces a functor between the associated exit path $\infty$-categories. As a consequence, we produce many new examples of exodromic stratified spaces, including: coarsenings of conical stratifications, locally finite subanalytic stratifications of real analytic spaces, and algebraic stratifications of real varieties. Our proofs are at the generality of stratified $\infty$-topoi, hence apply to even more general situations such as stratified topological stacks. In a subsequent paper, we use the previously mentioned finiteness results to construct derived moduli stacks of constructible and perverse sheaves.

math.AT↗

arXiv2026

Classifying anima of condensed $\infty$-categories of points

We compare the classifying anima of two natural condensed $\infty$-categories associated to a coherent $\infty$-topos. One from our work with Barwick and Glasman on exit-path categories in algebraic geometry, and the other from Lurie's work on ultracategories. The key consequence of our comparison is a connection between algebraic geometry and model theory: up to a mild completion, the proétale fundamental group of a scheme and the Lascar group of a complete first-order theory are both special cases of the same construction.

math.CT↗

arXiv2026

Localizing invariants of constructible sheaves

Given an open-closed decomposition of the stratifying poset, we construct a new semi-orthogonal decomposition of the $\infty$-category of constructible sheaves on a stratified space admitting an exit-path $\infty$-category. From this we obtain a direct sum decomposition of the localizing invariants of the $\infty$-category of constructible sheaves. Since the $\ast$-pullback to the open stratum in the usual (recollement) semi-orthogonal decomposition is not strongly left adjoint, this splitting does not follow from pure sheaf theory considerations. Instead, the splitting crucially relies on the exodromy equivalence: it implies that on the level of constructible sheaves, the $\ast$-pullback to a closed stratum and the $!$-pushforward from an open stratum admit left adjoints. These new functors provide an additional semi-orthogonal decomposition (with the roles of open and closed reversed) in which the relevant functors are strongly left adjoint.

math.KT↗

arXiv2025

Spectral weight filtrations

We provide a description of Voevodsky's $\infty$-category of motivic spectra in terms of the subcategory of motives of smooth proper varieties. As applications, we construct weight filtrations on the Betti and étale cohomologies of algebraic varieties with coefficients in any complex oriented ring spectrum. We show that these filtrations satisfy $\ell\mathrm{dh}$-descent, giving an effective way of calculating them in positive characteristic. In the complex motivic case, we further refine the weight filtration to one defined at the level of stable homotopy types.

math.AG↗

arXiv2025

The condensed homotopy type of a scheme

We study a condensed version of the étale homotopy type of a scheme, which refines both the usual étale homotopy type of Friedlander-Artin-Mazur and the proétale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a description of the condensed homotopy type using the Galois category of a scheme introduced by Barwick-Glasman-Haine. In the second part, we focus on the fundamental group of the condensed homotopy type in more detail. We show that, unexpectedly, the fundamental group of the condensed homotopy type of the affine line $\mathbf{A}^1_{\mathbf{C}}$ over the complex numbers is nontrivial. Nonetheless, its Noohi completion recovers the proétale fundamental group of Bhatt-Scholze. Moreover, we show that a mild correction, passing to the quasiseparated quotient, fixes most of this group's quirks. Surprisingly, this quotient is often a topological group.

math.AG↗

arXiv2025

Standard $t$-structures

We provide a general construction of induced $t$-structures, that generalizes standard $t$-structures for $\infty$-categories of sheaves. More precisely, given a presentable $\infty$-category $\mathcal{X}$ and a presentable stable $\infty$-category $\mathcal{E}$ equipped with an accessible $t$-structure $τ= (\mathcal{E}_{\geq 0}, \mathcal{E}_{\leq 0})$, we show that $\mathcal{X} \otimes \mathcal{E}$ is equipped with a canonical $t$-structure whose coconnective part is given in $\mathcal{X} \otimes \mathcal{E}_{\leq 0}$. When $\mathcal{X}$ is an $\infty$-topos, we give a more explicit description of the connective part as well.

math.CT↗

arXiv2025

Fully faithful functors and pushouts of $\infty$-categories

We study stability properties of fully faithful functors, and compute mapping anima in pushouts of $\infty$-categories along fully faithful functors. We provide applications of these calculations to pushouts along Dwyer functors and Reedy categories.

math.CT↗

arXiv2024

Reconstruction of schemes from their étale topoi

Let $k$ be a field that is finitely generated over its prime field. In Grothendieck's anabelian letter to Faltings, he conjectured that sending a $k$-scheme to its étale topos defines a fully faithful functor from the localization of the category of finite type $k$-schemes at the universal homeomorphisms to a category of topoi. We prove Grothendieck's conjecture for infinite fields of arbitrary characteristic. In characteristic $0$, this shows that seminormal finite type $k$-schemes can be reconstructed from their étale topoi, generalizing work of Voevodsky. In positive characteristic, this shows that perfections of finite type $k$-schemes can be reconstructed from their étale topoi.

math.AG↗

arXiv2024

Nonabelian basechange theorems & étale homotopy theory

This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in étale cohomology (i.e., prove analogues of the classical statements for sheaves of spaces). Second, we apply these theorems to prove a number of results about the étale homotopy type. Specifically, we prove nonabelian refinements of the smooth basechange theorem, Huber-Gabber affine analogue of the proper basechange theorem, and Fujiwara-Gabber rigidity theorem. Our methods also recover Chough's nonabelian refinement of the proper basechange theorem. Transporting an argument of Bhatt-Mathew to the nonabelian setting, we apply nonabelian proper basechange to show that the profinite étale homotopy type satisfies arc-descent. Using nonabelian smooth and proper basechange and descent, we give rather soft proofs of a number of Künneth formulas for the étale homotopy type.

math.AG↗

arXiv2024

Profinite completions of products

A source of difficulty in profinite homotopy theory is that the profinite completion functor does not preserve finite products. In this note, we provide a new, checkable criterion on prospaces $X$ and $Y$ that guarantees that the profinite completion of $X\times Y$ agrees with the product of the profinite completions of $X$ and $Y$. Using this criterion, we show that profinite completion preserves products of étale homotopy types of qcqs schemes. This fills a gap in Chough's proof of the Künneth formula for the étale homotopy type of a product of proper schemes over a separably closed field.

math.AT↗

arXiv2023

On the homotopy theory of stratified spaces

Let $P$ be a poset. We define a new homotopy theory of suitably nice $P$-stratified topological spaces with equivalences on strata and links inverted. We show that the exit-path construction of MacPherson, Treumann, and Lurie defines an equivalence from our homotopy theory of $P$-stratified topological spaces to the $\infty$-category of $\infty$-categories with a conservative functor to $P$. This proves a stratified form of Grothendieck's homotopy hypothesis, verifying a conjecture of Ayala-Francis-Rozenblyum. Our homotopy theory of stratified spaces has the added benefit of capturing all examples of geometric interest: conically stratified spaces fit into our theory, and the Ayala-Francis-Tanaka-Rozenblyum homotopy theory of conically smooth stratified spaces embeds into ours.

math.AT↗

arXiv2023

Differential Cohomology: Categories, Characteristic Classes, and Connections

We give an overview of differential cohomology from a modern, homotopy-theoretic perspective in terms of sheaves on manifolds. Although modern techniques are used, we base our discussion in the classical precursors to this modern approach, such as Chern-Weil theory and differential characters, and include the necessary background to increase accessibility. Special treatment is given to differential characteristic classes, including a differential lift of the first Pontryagin class. Multiple applications, including to configuration spaces, invertible field theories, and conformal immersions, are also discussed. This book is based on talks given at MIT's Juvitop seminar jointly with UT Austin in the Fall of 2019.

math.AT↗

arXiv2022

The fundamental fiber sequence in étale homotopy theory

Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences \begin{equation*} Π_{<\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to Π_{<\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \qquad \text{and} \qquad \widehatΠ{}_{\infty}^{\mathrm{\acute{e}t}}(X_{\bar{k}}) \to \widehatΠ{}_{\infty}^{\mathrm{\acute{e}t}}(X) \to \mathrm{BGal}(\bar{k}/k) \end{equation*} of protruncated and profinite étale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher étale homotopy groups of $X$ and the geometric fiber $X_{\bar{k}}$ are isomorphic, and second, if $X_{\bar{k}}$ is connected, then the sequence of profinite étale fundamental groups $1\to\hatπ{}_{1}^{\mathrm{\acute{e}t}}(X_{\bar{k}})\to\hatπ{}_{1}^{\mathrm{\acute{e}t}}(X)\to\mathrm{Gal}(\bar{k}/k)\to 1$ is exact. It also proves the analogous results for the `groupe fondamental élargi' of SGA3.

math.AT↗

arXiv2022

Descent for sheaves on compact Hausdorff spaces

These notes explain some descent results for $\infty$-categories of sheaves on compact Hausdorff spaces and derive some consequences. Specifically, given a compactly assembled $\infty$-category $\mathcal{E}$, we show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category $\operatorname{Sh}^{\operatorname{post}}(X;\mathcal{E})$ of Postnikov complete $\mathcal{E}$-valued sheaves on $ X $ satisfies descent for proper surjections. This implies proper descent for left complete derived $\infty$-categories and that the functor $\operatorname{Sh}^{\operatorname{post}}(-;\mathcal{E})$ is a sheaf on the category of compact Hausdorff spaces equipped with the topology of finite jointly surjective families. Using this, we explain how to embed Postnikov complete sheaves on a locally compact Hausdorff space into condensed objects. This implies that the condensed and sheaf cohomologies of a locally compact Hausdorff space agree.

math.AT↗

arXiv2022

From nonabelian basechange to basechange with coefficients

The goal of this paper is to explain when basechange theorems for sheaves of spaces imply basechange for sheaves with coefficients in other presentable $\infty$-categories. We accomplish this by analyzing when the tensor product of presentable $\infty$-categories preserves left adjointable squares. As a sample result, we show that the Proper Basechange Theorem in topology holds with coefficients in any presentable $\infty$-category which is compactly generated or stable. We also prove results about the interaction between tensor products of presentable $\infty$-categories and various categorical constructions that are of independent interest.

math.CT↗

arXiv2022

The homotopy-invariance of constructible sheaves

The purpose of this paper is to explain why the functor that sends a stratified topological space $S$ to the $\infty$-category of constructible (hyper)sheaves on $S$ with coefficients in a large class of presentable $\infty$categories is homotopy-invariant. To do this, we first establish a number of results in the unstratified setting, i.e., the setting of locally constant (hyper)sheaves. For example, if $X$ is a locally weakly contractible topological space and $\mathcal{E}$ is a presentable $\infty$-category, then we give a concrete formula for the constant hypersheaf functor $\mathcal{E}\to \mathrm{Sh}^{\mathrm{hyp}}(X;\mathcal{E})$. This formula lets us show that the constant hypersheaf functor is a right adjoint, and is fully faithful if $X$ is also weakly contractible. It also lets us prove a general monodromy equivalence and categorical Künneth formula for locally constant hypersheaves.

math.AT↗

arXiv2021

On the James and Hilton-Milnor Splittings, & the metastable EHP sequence

This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results \begin{equation*} ΣΩΣX \simeq ΣX \vee (X\wedge ΣΩΣX) \quad \text{and} \quad Ω(X \vee Y) \simeq ΩX\times ΩY\times ΩΣ(ΩX \wedge ΩY) \end{equation*} in the maximal generality of an $\infty$-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor Splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary $\infty$-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting result in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of $\infty$-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof.

math.AT↗