Proper Orbifold Cohomology

Abstract

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.