Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces
Abstract
The weak, stable, and ordinary Lusternik--Schnirelmann categories of a finite T0-space X satisfy catw(X)≤ cats(X)≤ cat(X). We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If P is weakly contractible but noncontractible and the deletion of one point makes P contractible, then adjoining m≥ 2 incomparable maximal points produces a connected finite space with category triple (1,2,m). Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on m+9 points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple (a,b,c) with 1≤ a<b≤ 2a and c≥ b, as well as every triple (a,a,c) with 2≤ a≤ c. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.
Related
- Lusternik-Schnirelmann category of Lee Forms on locally conformally symplectic manifolds
- Ordinary G-permutable subgroups of finite simple groups
- On the Stable category of maximal Cohen-Macaulay modules over Gorenstein rings-II
- Ordinary Nonconvex SGD under Distance-Dependent Moments: Finite-Horizon Stationarity and Nagaev Bounds
- Derived Ringel duality through finite subquotients of modular category 𝒪
BibTeX
@misc{mosqueralois2026weak,
title = {{Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces}},
author = {David Mosquera-Lois and Kohei Tanaka},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39615},
eprint = {2609.39615},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39615},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Weak, stable, and ordinary Lusternik-Schnirelmann category of finite spaces
AU - David Mosquera-Lois
AU - Kohei Tanaka
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39615
UR - https://arxiv.org/abs/2609.39615
AB - The weak, stable, and ordinary Lusternik--Schnirelmann categories of a finite T_0-space X satisfy cat_w(X)≤ cat_s(X)≤ cat(X). We give a general construction answering the simultaneous-strictness question of Cárdenas, Flores, Quintero, and Villar-Liñán. If P is weakly contractible but noncontractible and the deletion of one point makes P contractible, then adjoining m≥ 2 incomparable maximal points produces a connected finite space with category triple (1,2,m). Moreover, after any positive number of barycentric subdivisions its ordinary category is equal to $2$. Applying the construction to a nine-point space yields examples on m+9 points and, in particular, a twelve-point example for which both inequalities are strict. Using additivity under disjoint unions, we also realize every triple (a,b,c) with 1≤ a<b≤ 2a and c≥ b, as well as every triple (a,a,c) with 2≤ a≤ c. We conclude with questions concerning connected realizations and the minimum cardinality of a finite space exhibiting simultaneous strictness.
ER -