  The concept of orbifolds should unify differential geometry with equivariant
homotopy theory, so that orbifold cohomology should unify differential
cohomology with proper equivariant cohomology theory. Despite the prominent
role that orbifolds have come to play in mathematics and mathematical physics,
especially in string theory, the formulation of a general theory of orbifolds
reflecting this unification has remained an open problem. Here we present a
natural theory argued to achieve this. We give both a general abstract
axiomatization in higher topos theory, as well as concrete models for ordinary
as well as for super-geometric and for higher-geometric orbifolds. Our first
main result is a fully faithful embedding of the 2-category of orbifolds into a
singular-cohesive infinity-topos whose intrinsic cohomology theory is proper
globally equivariant differential generalized cohomology, subsuming traditional
orbifold cohomology, Chen-Ruan cohomology, and orbifold K-theory. Our second
main result is a general construction of orbifold etale cohomology which we
show to naturally unify (i) tangentially twisted cohomology of smooth but
curved spaces with (ii) RO-graded proper equivariant cohomology of flat but
singular spaces. As a fundamental example we present J-twisted orbifold
Cohomotopy theories with coefficients in shapes of generalized Tate spheres.
According to "Hypothesis H" this includes the proper orbifold cohomology theory
that controls non-perturbative string theory.
