Extending Hrushovski's groupoid-cover correspondence using simplicial groupoids

TL;DR

This paper extends Hrushovski's groupoid-cover correspondence by using simplicial groupoids to better understand 1-analysable covers in model theory, building on prior categorical and functorial frameworks.

Contribution

It introduces an extended equivalence of categories linking simplicial groupoids and 1-analysable covers, with new examples demonstrating the approach.

Findings

Established an equivalence of categories between simplicial groupoids and 1-analysable covers.

Extended previous work by Haykazyan and Moosa on groupoid-cover correspondence.

Provided illustrative examples applying the new theoretical framework.

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.