abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Brown-Comenetz 76 introduced the spectrum $I_{\mathbb{Q}/\mathbb{Z}}$ which represents the cohomology theory that assigns to a spectrum $X$ the algebraic Pontrjagin duals $Hom_{\mathbb{Z}}(\pi_{- \ast} X, \mathbb{Q}/\mathbb{Z})$ of its homotopy groups with respect to Q/Z.
If $X$ is a spectrum whose stable homotopy groups are finite groups, then the natural “double-duality” map $X \to I_{\mathbb{Q}/\mathbb{Z}} I_{\mathbb{Q}/\mathbb{Z}} X$ (the unit of the corresponding continuation monad) is an equivalence (see at dualizing object in a closed category).
The Brown-Comenetz spectrum $I_{\mathbb{Q}/\mathbb{Z}}$ has an “integral” refinement which is the Anderson spectrum $I_{\mathbb{Z}}$ inducing Anderson duality.
Last revised on December 18, 2020 at 17:35:23. See the history of this page for a list of all contributions to it.