Costabilisation of vₙ-Periodic Homotopy Types

One can consider the stabilisation of a symmetric monoidal ∞-category as the ∞-category of objects that admit an infinite delooping. For example, the ∞-category of spectra is the stabilisation of the ∞-category of homotopy types. Costabilisation is the opposite notion of stabilisation, where we are interested in objects that admits infinite desuspensions. It is easy to see that the costablisation of the ∞-category of homotopy types is trivial. The ∞-category of  vₙ-periodic homotopy types is a localisation of the ∞-category of homotopy types which is the analogue of rational localisation in higher chromatic height. In this work we showed that the costabilisation of vₙ-periodic homotopy types is the ∞-category of T(n)-local spectra. As a consequence, we obtain the universal property of the Bousfield–Kuhn functor. This is a joint work with Gijs Heuts.