Lie groupoids of mappings taking values in a Lie groupoid

Abstract

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper étale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current groupoid as a current algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators \[ C^\ell (K,f)\colon C^\ell (K,M)\rightarrow C^\ell (K,N)\,,\;\, \gamma f\circ \gamma \] between manifolds of $C^\ell $-functions. Under natural hypotheses, $C^\ell (K,f)$ turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map $f\colon M\rightarrow N$. These results are new in their generality and of independent interest.