Connected Lie Groupoids are Internally Connected and Integral Complete in Synthetic Differential Geometry

TL;DR

This paper reformulates key concepts of Lie groupoids within synthetic differential geometry, demonstrating that classical connectedness conditions imply their internal analogues in this framework.

Contribution

It introduces internal definitions of source path and simply connected groupoids and proves their equivalence to classical notions under certain conditions.

Findings

Classical connectedness implies internal connectedness in synthetic differential geometry.

Internal source path and simply connected groupoids are well-defined and consistent.

The work bridges classical Lie theory and synthetic differential geometry.

Abstract

We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of $A$-paths. The main results of this paper show that if a classical Hausdorff Lie groupoid satisfies one of the classical connectedness conditions it also satisfies its internal counterpart.