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Groupoids, imaginaries and internal covers

Ehud Hrushovski

Abstract

Let $T$ be a first-order theory. A correspondence is established between internal covers of models of $T$ and definable groupoids within $T$. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between: covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of $T^\si$, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.

Groupoids, imaginaries and internal covers

Abstract

Let be a first-order theory. A correspondence is established between internal covers of models of and definable groupoids within . We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between: covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of , and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.

Paper Structure

This paper contains 7 sections, 24 theorems, 15 equations.

Table of Contents

  1. Preliminaries
  2. Definable groupoids
  3. Generalized imaginaries
  4. Higher amalgamation
  5. Adding an automorphism
  6. Linear imaginaries
  7. Pseudo-finite fields

Key Result

Lemma 1.3

Let $T'$ be a theory, $T$ the restriction of $T'$ to a subset of the sorts of $T'$, $T"$ an expansion of $T'$ on the same sorts as $T'$. Assume $T$ is stably embedded in $T'$, and for any $N" \models T"$, if $N',N$ are the restrictions to $T',T$ respectively, the natural map $Aut(N"/N) \to Aut(N'/N)

Theorems & Definitions (47)

  • Definition 1.1
  • Remark 1.2
  • Lemma 1.3
  • proof
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • ...and 37 more