David Roberts (University of Adelaide, Australia) Smooth Mapping Stacks of Differentiable Stacks and Orbifolds Sponsored by the Meyer Fund
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The concept of the space of all maps of a particular kind between a pair of manifolds is an old one, and it is the subject of global analysis to study the structure of such spaces and when one can do differential geometry on them. The nicest possible example of such a space is the (infinite-dimensional) Fréchet manifold of smooth loops in a smooth manifold $M$. More generally, given a compact smooth manifold $K$ and a smooth manifold $M$, one can define the Fréchet manifold of smooth maps from $K$ to $M$. The assumption of compactness of $K$ is not necessary, but then the topology gets a lot more wild. Moving just beyond manifolds, one can consider the space of smooth loops in an orbifold, which is important to understanding the string topology of orbifolds, or more generally, what a smooth mapping space from a compact manifold to an orbifold looks like. Such a thing will not necessarily be a priori a smooth manifold. Despite several attempts in the literature, the constructions have been extremely messy, and at least one case has counterexamples to some of its claims. The most mature approach to orbifolds treats them as differentiable stacks with the property that they are presented by a proper étale Lie groupoid. This talk will outline a recent construction of a presentation by a Lie groupoid of the smooth mapping stack of maps from a compact manifold (possibly with corners) to a differentiable stack $X$. In the special case that $X$ is an orbifold, the mapping stack is then presented by a proper étale Fréchet-Lie groupoid, a candidate for an infinite-dimensional smooth orbifold.