The stratified structure of spaces of smooth orbifold mappings

We consider four notions of maps between smooth C^∞ orbifolds $\mathcal{O}$, $\mathcal{P}$ with $\mathcal{O}$ compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of C^r maps between $\mathcal{O}$ and $\mathcal{P}$ with the C^r topology carries the structure of a smooth C^∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of C^r maps between $\mathcal{O}$ and $\mathcal{P}$ with the C^r topology carries the structure of a smooth C^∞ Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The C^r orbifold maps between $\mathcal{O}$ and $\mathcal{P}$ is locally a stratified space with strata modeled on smooth C^∞ Banach (r finite)/Fréchet (r = ∞) manifolds while the set of C^r reduced orbifold maps between $\mathcal{O}$ and $\mathcal{P}$ locally has the structure of a stratified space with strata modeled on smooth C ^∞ Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C^∞ Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.