The Oriented Freudenthal Suspension Theorem

The classical Freudenthal Suspension Theorem is the statement that, given a space X, the connectivity of the unit map X -> ΩΣX of the loop-suspension adjunction is (roughly) twice the connectivity of X. In this talk, I will discuss the setting of oriented (or directed) homotopy theory, in which spaces are replaced by higher, i.e. (∞,ω)-, categories and the cartesian product by the Gray tensor product, and generalize the Freudenthal Suspension Theorem in this context. This gives, in particular, evidence for a yet-elusive "oriented topos theory".