Pre-Galois categories and Fra\"iss\'e's theorem

TL;DR

This paper introduces pre-Galois categories as combinatorial analogs of pre-Tannakian categories, showing that categories of finitary smooth G-sets for oligomorphic groups exhaust these structures, reformulating Fra"issé's theorem.

Contribution

It defines pre-Galois categories, proves their exhaustiveness via a reformulation of Fra"issé's theorem, and introduces B-categories with examples beyond pre-Galois.

Findings

Categories of finitary smooth G-sets are pre-Galois.

Main theorem characterizes all pre-Galois categories as such G-set categories.

Examples of B-categories not pre-Galois are provided.

Abstract

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\"iss\'e'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.