Atomic toposes, choice spectra and model-theoretic stability
Abstract
For a suitable class of structures 𝒦, let Sh(𝒦λop,Jat) denote the atomic sheaf topos over the opposite of the category 𝒦λ of structures in 𝒦 of cardinality at most λ, with embeddings as morphisms. In this article, we analyze the relationship between model-theoretic properties of 𝒦 and logical properties of Sh(𝒦λop,Jat). For an infinite cardinal λ, we show that 𝒦 is Galois λ-stable if and only if the atomic sheaf topos Sh(𝒦λop,Jat) satisfies the axiom of choice for λ+-indexed families. We further show that every structure in 𝒦λ extends to a maximal structure if and only if Sh(𝒦λop,Jat) satisfies the internal axiom of choice. If 𝒦fin is a class of finite structures, the cardinalities of the structures in each connected component of 𝒦fin have a finite upper bound if and only if Sh(𝒦opfin,Jat) satisfies the internal axiom of (countable) choice. Furthermore, via the topological Galois representation, we establish a correspondence between the criteria for the axiom of choice for atomic sheaf toposes and those for permutation models in set theory.
Related
- Coefficient choice in Galois-ring lifts of APN permutations: sharp unit-direction spectra and one-term optimal lifts
- Operator-Theoretic Stability and Observer Synthesis for Parameter-Dependent Vlasov--Maxwell Dynamics
- On splitting properties of the stability problem with integer choice functions
- Grove polynomials and K-theoretic quasisymmetry
- Conservative CSPs in Logspace are in Symmetric Linear Datalog (under complexity theoretic assumptions)
BibTeX
@misc{kihara2026atomic,
title = {{Atomic toposes, choice spectra and model-theoretic stability}},
author = {Takayuki Kihara},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21111},
eprint = {2609.21111},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21111},
}
RIS
TY - UNPB
TI - Atomic toposes, choice spectra and model-theoretic stability
AU - Takayuki Kihara
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21111
UR - https://arxiv.org/abs/2609.21111
AB - For a suitable class of structures 𝒦, let Sh(𝒦_λ^op,J_at) denote the atomic sheaf topos over the opposite of the category 𝒦_λ of structures in 𝒦 of cardinality at most λ, with embeddings as morphisms. In this article, we analyze the relationship between model-theoretic properties of 𝒦 and logical properties of Sh(𝒦_λ^op,J_at). For an infinite cardinal λ, we show that 𝒦 is Galois λ-stable if and only if the atomic sheaf topos Sh(𝒦_λ^op,J_at) satisfies the axiom of choice for λ^+-indexed families. We further show that every structure in 𝒦_λ extends to a maximal structure if and only if Sh(𝒦_λ^op,J_at) satisfies the internal axiom of choice. If 𝒦_fin is a class of finite structures, the cardinalities of the structures in each connected component of 𝒦_fin have a finite upper bound if and only if Sh(𝒦^op_fin,J_at) satisfies the internal axiom of (countable) choice. Furthermore, via the topological Galois representation, we establish a correspondence between the criteria for the axiom of choice for atomic sheaf toposes and those for permutation models in set theory.
ER -