The smooth Hom-stack of an orbifold

TL;DR

This paper demonstrates that the Hom-stack of maps from a compact manifold to an orbifold can be represented as an infinite-dimensional orbifold, providing a new perspective on the structure of mapping stacks in differential geometry.

Contribution

It establishes that the Hom-stack from a compact manifold to an orbifold is itself an infinite-dimensional orbifold, presented by a proper étale Fréchet-Lie groupoid.

Findings

Hom(M,X) is presented by a Fre9chet-Lie groupoid Map(M,X)

If X is an orbifold, Map(M,X) is proper étale

Hom-stack is an infinite-dimensional differentiable stack

Abstract

For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fr\'echet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper \'etale Lie groupoid, then Map(M,X) is proper \'etale and so presents an infinite-dimensional orbifold.