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Pre-Galois categories and Fraïssé's theorem

Nate Harman, Andrew Snowden

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 Fraïssé'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.

Pre-Galois categories and Fraïssé's theorem

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 , the category of finitary smooth -sets is pre-Galois. Our main theorem (approximately) says that these examples are exhaustive; this result is, in a sense, a reformulation of Fraïssé'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.
Paper Structure (42 sections, 51 theorems, 16 equations)

This paper contains 42 sections, 51 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Pre-Galois categories
  4. Examples
  5. The main theorem
  6. Overview of proof
  7. Motivation
  8. Outline
  9. Notation
  10. Oligomorphic groups
  11. Oligomorphic groups
  12. Admissible groups
  13. Actions
  14. Combinatorial tensor categories
  15. Basic definitions
  16. ...and 27 more sections

Key Result

Theorem 1.2

If $\mathcal{B}$ is a pre-Galois category then $\mathcal{B}$ is equivalent to $\mathbf{S}(G)$ for some admissible group $G$.

Theorems & Definitions (119)

  • Definition 1.1
  • Theorem 1.2: Theorem \ref{['thm:uncountable']}
  • Remark 1.3
  • Example 2.1
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 109 more