Sheaves on polynomials and polynomial sheaves
Abstract
We show that several Dold-Kan type equivalences can be understood as a passage from constructible sheaves on symmetric products (i.e. spaces of polynomials) to polynomial sheaves in the sense of Goodwillie-Weiss calculus, by means of a natural analytic expansion functor. This geometric perspective provides access to standard tools from sheaf theory such as the six functor formalism and t-structures. For example, the equivalence between filtered objects and coherent chain complexes in a stable ∞-category receives a geometric incarnation as Verdier duality between !-sheaves and *-cosheaves on an ∞-dimensional simplex. As a main application, we obtain an intrinsic geometric explanation for the appearance of vanishing cycle data for perverse sheaves in a paracyclic variant of the Dold--Kan correspondence pioneered by Dwyer and Kan, at the same time lifting their equivalence to stable ∞-categories.
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BibTeX
@misc{dyckerhoff2026sheaves,
title = {{Sheaves on polynomials and polynomial sheaves}},
author = {Tobias Dyckerhoff and Till Heine and Simon Schneider},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36314},
eprint = {2610.36314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36314},
}
RIS
TY - UNPB
TI - Sheaves on polynomials and polynomial sheaves
AU - Tobias Dyckerhoff
AU - Till Heine
AU - Simon Schneider
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36314
UR - https://arxiv.org/abs/2610.36314
AB - We show that several Dold-Kan type equivalences can be understood as a passage from constructible sheaves on symmetric products (i.e. spaces of polynomials) to polynomial sheaves in the sense of Goodwillie-Weiss calculus, by means of a natural analytic expansion functor. This geometric perspective provides access to standard tools from sheaf theory such as the six functor formalism and t-structures. For example, the equivalence between filtered objects and coherent chain complexes in a stable ∞-category receives a geometric incarnation as Verdier duality between !-sheaves and *-cosheaves on an ∞-dimensional simplex. As a main application, we obtain an intrinsic geometric explanation for the appearance of vanishing cycle data for perverse sheaves in a paracyclic variant of the Dold--Kan correspondence pioneered by Dwyer and Kan, at the same time lifting their equivalence to stable ∞-categories.
ER -