Reconstruction of graphs and surfaces from derived categories
Abstract
A pair of topological spaces X and Y is a \emph{Fourier--Mukai (FM) pair} if there is an equivalence of ∞-categories of derived sheaves of abelian groups Shv(X;D(ℤ))≃ Shv(Y;D(ℤ)). An FM pair (X,Y) is \emph{non-trivial} if X and Y are not homeomorphic. We consider the question of whether non-trivial FM pairs (X,Y) exist for which X and Y are both "nice" spaces, which in this article means admitting a triangulation. We show that the answer is no if X is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of \emph{Tate--Hochschild cohomology} (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold X, we relate the Tate--Hochschild cohomology of Shv(X;D(ℤ)) to the homology at infinity of X (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory 𝒟Gor⊆𝒟 called the \emph{Gorenstein locus}, defined for every dualizable stable category 𝒟. For a triangularizable space X and a field k, we show that the Gorenstein locus of Shv(X;D(k)) is equal to the full subcategory of sheaves supported on the k-homology manifold locus of X.
Related
- The derived Picard group of Fukaya categories of closed surfaces
- Derived Reconstruction of Semisimple Tensor and Module Categories
- Derived categories of log del Pezzo surfaces with 1/3(1,1) singularities
- Derived Picard reconstruction for graded surfaces with stops
- Fukaya categories of surfaces with DG coefficients
BibTeX
@misc{harr2026reconstruction,
title = {{Reconstruction of graphs and surfaces from derived categories}},
author = {Oscar Harr},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13488},
eprint = {2610.13488},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13488},
}
RIS
TY - UNPB
TI - Reconstruction of graphs and surfaces from derived categories
AU - Oscar Harr
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13488
UR - https://arxiv.org/abs/2610.13488
AB - A pair of topological spaces X and Y is a \emph{Fourier--Mukai (FM) pair} if there is an equivalence of ∞-categories of derived sheaves of abelian groups Shv(X;D(ℤ))≃ Shv(Y;D(ℤ)). An FM pair (X,Y) is \emph{non-trivial} if X and Y are not homeomorphic. We consider the question of whether non-trivial FM pairs (X,Y) exist for which X and Y are both "nice" spaces, which in this article means admitting a triangulation. We show that the answer is no if X is second-countable and either (i) one-dimensional, (ii) a surface, or (iii) the 3-sphere. In order to prove this, we define two invariants of dualizable categories. The first invariant is an extension of \emph{Tate--Hochschild cohomology} (aka singular Hochschild cohomology) to a class of dualizable categories satisfying a countability assumption. For a homology manifold X, we relate the Tate--Hochschild cohomology of Shv(X;D(ℤ)) to the homology at infinity of X (in the sense of proper homotopy theory). The second invariant takes the form of a subcategory 𝒟_Gor⊆𝒟 called the \emph{Gorenstein locus}, defined for every dualizable stable category 𝒟. For a triangularizable space X and a field k, we show that the Gorenstein locus of Shv(X;D(k)) is equal to the full subcategory of sheaves supported on the k-homology manifold locus of X.
ER -