Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.13956

Homological stratification and descent

Tobias Barthel·Drew Heard·Beren Sanders·Changhan Zou
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a notion of stratification for rigidly-compactly generated tensor-triangulated categories relative to the homological spectrum and develop the fundamental features of this theory. In particular, we demonstrate that it exhibits excellent descent properties. In conjunction with Balmer's Nerves of Steel conjecture, we conclude that stratification admits a general form of descent. This gives a uniform treatment of several recent stratification results and provides a complete answer to the question: When does stratification descend? As a new application, we extend earlier work on the tensor triangular geometry of equivariant module spectra from finite groups to compact Lie groups.

Explore related subjects

math.CTmath.ATmath.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tobias Barthel, Drew Heard, Beren Sanders, Changhan Zou. 2026-03-18. Homological stratification and descent. https://doi.org/10.1017/s1474748025101515

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hochschild cohomology of the second kind: Koszul duality and Morita invariance

We define Hochschild cohomology of the second kind for differential graded (dg) or curved algebras as a derived functor in the twisted derived category. The Hochschild cohomology of the second kind of a curved or curved algebra $A$ is then equivalent to the classical Hochschild cohomology of the twisted derived dg category of $A$, which is often geometrically meaningful. Examples include the category of $\infty$-local systems on a topological space, the bounded derived category of a complex manifold and the category of matrix factorizations. We also show that Hochschild cohomology of the second kind is preserved under (nonconilpotent) Koszul duality and weak equivalences of curved algebras. The main technical ingredient is a new bimodule version of Koszul duality.

math.CT↗
arXiv2026

Localization of lax symmetric monoidal categories

In this note, we explain in some detail how one can fiberwise localize a (co)lax symmetric monoidal infinity-category. This construction was tacitly used in Section 5 of our recent paper "On the equivalence of the Lurie's infinity-operads and dendroidal infinity-operads". Version 2: A stronger version of the result is proven. Given a locally cocartesian fibration $f:X\to B$ and a collection of marked arrows $X^\circ\subset f^{-1}(B^{eq})$ in $X$ closed under locally cocartesian liftings, we prove that the localization $\mathcal{L}(X,X^\circ)\to B$ is also a locally cocartesian fibration whose fibers are localizations of the fibers of $f$. This result is applied to the description of localizations of lax symmetric monoidal categories.

math.CT↗
arXiv2026

Cocompactness and Presentability

We give a short proof that $κ$-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable $\infty$-categories. A consequence is that an $\infty$-category $\mathcal{C}$ such that both $\mathcal{C}$ and $\mathcal{C}^\mathsf{op}$ are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable $\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.

math.CT↗