A Differentiable Monoid of Smooth Maps on Lie Groupoids

We investigate a monoid of smooth mappings on the space of arrows of a Lie groupoid and its group of units. The group of units turns out to be an infinite-dimensional Lie group which is regular in the sense of Milnor. Furthermore, this group is closely connected to the group of bisections and the geometry of the Lie groupoid. Under suitable conditions, i.e. if the source map of the Lie groupoid is proper, one also obtains a differentiable structure on the monoid and can identify the bisection group as a Lie subgroup of its group of units. Finally, relations between the (sub-)groupoids associated to the underlying Lie groupoid and subgroups of the monoid are obtained. The key tool driving the investigation is a generalisation of a result by A. Stacey. In the present article, we establish this so-called Stacey-Roberts Lemma. It asserts that pushforwards of submersions are submersions between the infinite-dimensional manifolds of mappings. The Stacey-Roberts Lemma is of independent interest as it provides tools to study submanifolds of and geometry on manifolds of mappings.