arXiv subjects

Alice Hedenlund

Publications and source records attributed to Alice Hedenlund.

5 recordsLinked to original sources

arXiv2026

Structured flow categories and twisted presheaves

An orientation theory for flow categories without bubbling is determined by a functor of $\infty$-categories $\mu \colon \mathcal{C} \to U/O$. For any such functor, we construct a stable $\infty$-category $\mathcal{F}low^{\mu}$ of $\mu$-structured flow categories and bimodules. We also construct the expected functors between such $\infty$-categories, giving a tractable framework for manipulating orientations, local systems, and filtrations in exact Floer homotopy theory. Classifying spaces for certain bordism theories determined by $\mu$ appear as mapping spaces in $\mathcal{F}low^{\mu}$, and we use a Pontrjagin--Thom construction to naturally identify $\mathcal{F}low^{\mu}$ with the $\infty$-category of $\mu$-twisted presheaves on $\mathcal{C}$.

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arXiv2024

The Synthetic Hilbert Additive Group Scheme

We construct a lift of the degree filtration on the integer valued polynomials to (even MU-based) synthetic spectra. Namely, we construct a bialgebra in modules over the evenly filtered sphere spectrum which base-changes to the degree filtration on the integer valued polynomials. As a consequence, we may lift the Hilbert additive group scheme to a spectral group scheme over $\mathbb{A}^1/\mathbb{G}_m$. We study the cohomology of its deloopings, and show that one obtains a lift of the filtered circle, studied in [MRT22]. At the level of quasi-coherent sheaves, one obtains lifts synthetic lifts of the $\mathbb{Z}$-linear $\infty$-categories of $S^1_{\mathrm{fil}}$-representations. Our constructions crucially rely on the use of the even filtration of Hahn--Raksit--Wilson; it is linearity with respect to the even filtered sphere that powers the results of this work.

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arXiv2023

Twisted Spectra Revisited

We recapture Douglas' framework for twisted parametrized stable homotopy theory in the language of $\infty$- categories. A twisted spectrum is essentially a section of a bundle of presentable stable $\infty$-categories whose fiber is the $\infty$-category of spectra, a perspective we refine in this work. We recover some of Douglas' results on classifications of such bundles, as well as further make precise the identification of categories of twisted spectra with module categories over Thom spectra in the pointed connected case. Furthermore, we examine subtle aspects of the functoriality which arise by virtue of being fibered over the Brauer space of the sphere spectrum. Extending beyond the scope of Douglas's work, we introduce a total category of twisted spectra over a fixed space, where we allow the twists to vary, and show that these total categories themselves satisfy the essential features of a 6-functor formalism. We also introduce an $(\infty, 2)$-category of twisted spectra, and use this framework to discuss duality theory for these type of objects.

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arXiv2021

A multiplicative Tate spectral sequence for compact Lie group actions

Given a compact Lie group $G$ and a commutative orthogonal ring spectrum $R$ such that $R[G]_* = π_*(R \wedge G_+)$ is finitely generated and projective over $π_*(R)$, we construct a multiplicative $G$-Tate spectral sequence for each $R$-module $X$ in orthogonal $G$-spectra, with $E^2$-page given by the Hopf algebra Tate cohomology of $R[G]_*$ with coefficients in $π_*(X)$. Under mild hypotheses, such as $X$ being bounded below and the derived page $RE^\infty$ vanishing, this spectral sequence converges strongly to the homotopy $π_*(X^{tG})$ of the $G$-Tate construction $X^{tG} = [\widetilde{EG} \wedge F(EG_+, X)]^G$.

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arXiv2020

Loday constructions on twisted products and on tori

We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted products in the case where the group involved is a constant simplicial group. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung. We prove that several truncated polynomial rings are not multiplicatively stable by investigating their torus homology.

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