Pre-Galois categories and Fraïssé's theorem

Galois categories can be viewed as the combinatorial analog of Tannakian categories. We introduce the notion of pre-Galois category, which can be viewed as the combinatorial analog of pre-Tannakian categories. Given an oligomorphic group \(G\), the category \(\mathbf{S}(G)\) of finitary smooth \(G\)...

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Bibliographic Details
Published in:arXiv.org 2024-02
Main Authors: Harman, Nate, Snowden, Andrew
Format: Article
Language:English
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Combinatorial analysis
Permutations
Theorems
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Summary:Galois categories can be viewed as the combinatorial analog of Tannakian categories. We introduce the notion of pre-Galois category, which can be viewed as the combinatorial analog of pre-Tannakian categories. Given an oligomorphic group \(G\), the category \(\mathbf{S}(G)\) of finitary smooth \(G\)-sets is pre-Galois. Our main theorem (approximately) says that these examples are exhaustive; this result is, in a sense, a reformulation of Fra\"issé's theorem. We also introduce a more general class of B-categories, and give some examples of B-categories that are not pre-Galois using permutation classes. This work is motivated by certain applications to pre-Tannakian categories.
ISSN:2331-8422