### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 007, 25 pages      arXiv:1606.06120      https://doi.org/10.3842/SIGMA.2017.007

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

Matthew Burke
4 River Court, Ferry Lane, Cambridge CB4 1NU, UK

Received June 29, 2016, in final form January 13, 2017; Published online January 24, 2017; Statement of Theorem 4.7 and notation in Section 4.3 corrected April 14, 2017

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.

Key words: Lie theory; Lie groupoid; Lie algebroid; category theory; synthetic differential geometry; intuitionistic logic.

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