Functoriality and uniformity in Hrushovski's groupoid-cover correspondence

TL;DR

This paper extends Hrushovski's correspondence between definable groupoids and internal imaginaries, proving it is an equivalence of categories that varies uniformly in definable families, with additional elaborations on the original constructions.

Contribution

It establishes that the correspondence is an equivalence of categories and demonstrates its uniform variation in definable families, enhancing the original framework.

Findings

The correspondence is an equivalence of categories.

The equivalence varies uniformly in definable families.

Additional elaborations on Hrushovski's constructions are provided.

Abstract

The correspondence between definable connected groupoids in a theory $T$ and internal generalised imaginary sorts of $T$, established by Hrushovski in ["Groupoids, imaginaries and internal covers," Turkish Journal of Mathematics, 2012], is here extended in two ways: First, it is shown that the correspondence is in fact an equivalence of categories, with respect to appropriate notions of morphism. Secondly, the equivalence of categories is shown to vary uniformly in definable families, with respect to an appropriate relativisation of these categories. Some elaboration on Hrushovki's original constructions are also included.