Orbifold bordism and duality for finite orbispectra

We construct the stable (representable) homotopy category of finite orbispectra, whose objects are formal desuspensions orbi-CW-pairs by vector bundles and morphisms classes relative maps. The representable orbispectra admits a contravariant involution extending Spanier--Whitehead duality. This duality relates homotopical cobordism theories (cohomology on orbispectra) represented global Thom spectra to geometric (derived) orbifold bordism groups (homology orbispectra). isomorphism extends classical Pontryagin--Thom its known equivariant generalizations.