Online Talk

We all know that only finite-dimensional vector spaces are reflexive, i.e. map isomorphically to their double dual. However, if one remembers a topology on the dual space and uses continuous duals, reflexivity can hold for wider classes of vector spaces. One way to encode a topology is to use a condensed structure, i.e. work in the topos of (kappa-)condensed sets. However, while this repairs reflexivity for some vector spaces, for more complicated vector spaces issues persist.

One can then more broadly ask whether there is any topos in which all vector spaces are reflexive. An easy argument shows that this must fail at the abelian level; however, it does not preclude reflexivity at the level of the derived category of vector spaces. It seems very hard to construct such a topos directly, but in joint work with Stefanich, we happened to accidentally run into an example that supports such reflexivity on not all, but an extremely broad class of vector spaces: This is the topos of Gestalten. The reflexivity comes from an extremely general form of Cartier duality in Gestalten, also related to the classification of invertible topological quantum field theories and the Brown-Comenetz dual of the sphere spectrum.