Locally definable homotopy

TL;DR

This paper extends o-minimal homotopy theory to locally definable spaces, introduces homology and homotopy functors for these spaces, and analyzes connectedness concepts in V-definable groups.

Contribution

It generalizes o-minimal homotopy results to locally definable spaces and studies connectedness notions in V-definable groups.

Findings

Extended homotopy and homology theories to locally definable spaces.

Proved non-equivalence of connectedness concepts in V-definable groups.

Established foundational results for locally definable homotopy theory.

Abstract

In "On o-minimal homotopy groups", o-minimal homotopy was developed for the definable category, proving o-minimal versions of the Hurewicz theorems and the Whitehead theorem. Here, we extend these results to the category of locally definable spaces, for which we introduce homology and homotopy functors. We also study the concept of connectedness in V-definable groups -- which are examples of locally definable spaces. We show that the various concepts of connectedness associated to these groups, which have appeared in the literature, are non-equivalent.