Gabriel Spectrum of Persistence Categories
Abstract
We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space X, we prove that the Gabriel spectrum of Sh(X) is homeomorphic to Sk(Sob(X)), where Sob(X) denotes the sobrification of X and Sk indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if X is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ℝn is homeomorphic to the space of ideals of ℝn with a natural topology.
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BibTeX
@misc{harsu2026gabriel,
title = {{Gabriel Spectrum of Persistence Categories}},
author = {Manu Harsu and Eero Hyry},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25761},
eprint = {2609.25761},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25761},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gabriel Spectrum of Persistence Categories
AU - Manu Harsu
AU - Eero Hyry
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25761
UR - https://arxiv.org/abs/2609.25761
AB - We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space X, we prove that the Gabriel spectrum of Sh(X) is homeomorphic to Sk(Sob(X)), where Sob(X) denotes the sobrification of X and Sk indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if X is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ℝ^n is homeomorphic to the space of ideals of ℝ^n with a natural topology.
ER -