Lax descent and foliated Verdier duality
Abstract
Higher categorical methods allow us to view recollements as lax descent data and build a corresponding descent theory governed by an ∞-bicategorical theory of limits. We demonstrate some concrete innovations brought along by these new techniques in the classical context of constructible and perverse sheaves on stratified spaces. Our first application uses partially (op)lax descent to recover the famous result of Beilinson telling us how to glue perverse sheaves in terms of vanishing and nearby cycles. The remainder of the article is concerened with establishing a uniform view on a series of classification results for perverse sheaves culminating in recent work of Kapranov-Schechtman. Namely, we use lax descent to establish a ``foliated'' variant of Verdier duality which, informally speaking, identifies the derived constructible category of a suitably foliated space with sections of a categorical cosheaf on the leaf space whose stalks are the categories of sheaves along the leaves. Again, this is obtained as an application of lax descent, and seems to provide a reasonable framework to formalize the idea of ``mixed sheaf-cosheaf objects'' promoted by Kapranov and Schechtman.
Related
- Euler obstructions and Verdier specializations
- Verdier-Type Category, Geometric Serre's conditions and perversity
- Ranks of Verdier quotients of derived categories over local rings
- Comparing metric completions of triangulated categories with Verdier quotients
- On deformations of foliated complex analytic structures II: Functors of Artin rings
BibTeX
@misc{dyckerhoff2026descent,
title = {{Lax descent and foliated Verdier duality}},
author = {Tobias Dyckerhoff and Till Heine and Simon Schneider and Kostiantyn Tolmachov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29977},
eprint = {2610.29977},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29977},
}
RIS
TY - UNPB
TI - Lax descent and foliated Verdier duality
AU - Tobias Dyckerhoff
AU - Till Heine
AU - Simon Schneider
AU - Kostiantyn Tolmachov
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29977
UR - https://arxiv.org/abs/2610.29977
AB - Higher categorical methods allow us to view recollements as lax descent data and build a corresponding descent theory governed by an ∞-bicategorical theory of limits. We demonstrate some concrete innovations brought along by these new techniques in the classical context of constructible and perverse sheaves on stratified spaces. Our first application uses partially (op)lax descent to recover the famous result of Beilinson telling us how to glue perverse sheaves in terms of vanishing and nearby cycles. The remainder of the article is concerened with establishing a uniform view on a series of classification results for perverse sheaves culminating in recent work of Kapranov-Schechtman. Namely, we use lax descent to establish a ``foliated'' variant of Verdier duality which, informally speaking, identifies the derived constructible category of a suitably foliated space with sections of a categorical cosheaf on the leaf space whose stalks are the categories of sheaves along the leaves. Again, this is obtained as an application of lax descent, and seems to provide a reasonable framework to formalize the idea of ``mixed sheaf-cosheaf objects'' promoted by Kapranov and Schechtman.
ER -