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Extending Hrushovski's groupoid-cover correspondence using simplicial groupoids

Paul Z. Wang

TL;DR

The ideas of Haykazyan and Moosa, found in ["Functoriality and uniformity in Hrushovski's groupoid-cover correspondence," Annals of Pure and Applied Logic], are used, and extended, to define an equivalence of categories.

Abstract

Hrushovski's suggestion, given in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics , 2012], to capture the structure of the 1-analysable covers of a theory T using simplicial groupoids definable in T is realized here. The ideas of Haykazyan and Moosa, found in ["Functoriality and uniformity in Hrushovski's groupoid-cover correspondence," Annals of Pure and Applied Logic , 2018] are used, and extended, to define an equivalence of categories. Finally, a couple of examples are studied with these new tools.

Extending Hrushovski's groupoid-cover correspondence using simplicial groupoids

TL;DR

The ideas of Haykazyan and Moosa, found in ["Functoriality and uniformity in Hrushovski's groupoid-cover correspondence," Annals of Pure and Applied Logic], are used, and extended, to define an equivalence of categories.

Abstract

Hrushovski's suggestion, given in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics , 2012], to capture the structure of the 1-analysable covers of a theory T using simplicial groupoids definable in T is realized here. The ideas of Haykazyan and Moosa, found in ["Functoriality and uniformity in Hrushovski's groupoid-cover correspondence," Annals of Pure and Applied Logic , 2018] are used, and extended, to define an equivalence of categories. Finally, a couple of examples are studied with these new tools.

Paper Structure

This paper contains 28 sections, 68 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Internal covers and 1-analysable covers with independent fibers
  3. General definitions and context
  4. Connected groupoids
  5. Non-connected groupoids
  6. Functoriality in the independent case
  7. Categories of groupoids, categories of covers
  8. The functor $C : \mathrm{FCG_A} \rightarrow \mathrm{ACIF_A}$
  9. The functor $G :\mathrm{ACIF_A} \rightarrow \mathrm{FCG_A}$
  10. The isomorphism $\eta : id_{ACIF_A} \simeq CG$
  11. The isomorphism $\varepsilon : id_{FCG_A} \simeq GC$
  12. Analysable covers and simplicial groupoids
  13. Constructing a simplicial groupoid from a 1-analysable cover
  14. Constructing a 1-analysable cover from a simplicial groupoid
  15. Comparing covers
  16. ...and 13 more sections

Key Result

Proposition 2.2

Let $T'$ be a complete extension of $T$, possibly with additional sorts. Let $\mathbb{U}'$ be a saturated model of $T'$, and $\mathbb{U}:= \mathbb{U'}|_T$. The following are equivalent :

Theorems & Definitions (176)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Theorem 2.9: hrushovski, haykazyan-moosa
  • proof
  • ...and 166 more