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Seminario del 2024
Venerdì
15 novembre
Matthew Di Meglio
nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR
Seminario di algebra e geometria, interdisciplinare, logica, teoria delle categorie
ore
14:00
presso Seminario II
The notion of abelian category is an elegant distillation of the fundamental properties of the category of abelian groups, comprising a few simple axioms about products and kernels. While the categories of real and complex Hilbert spaces and bounded linear maps are not abelian, they satisfy almost all of the abelian category axioms. Heunen and Kornell’s recent characterisation (https://doi.org/10.1073/pnas.2117024119) of these categories of Hilbert spaces is reminiscent of the Freyd–Mitchell embedding theorem, which says that every abelian category has a full, faithful and exact embedding into the category of modules over a ring. The axioms are similar, but incorporate the extra structure of a dagger—an identity-on-objects involutive contravariant endofunctor—which encodes adjoints of bounded linear maps. By keeping only the axioms that directly parallel the ones for abelian categories, we arrive at a nice class of dagger categories, which I call rational dagger categories, that enjoy many of the same properties as the categories of Hilbert spaces mentioned above. The name alludes to their unique enrichment in the category of rational vector spaces.
In this talk, I will give a gentle introduction to rational dagger categories, highlighting the parallels with abelian categories. I will not assume prior familiarity with dagger categories, instead introducing the relevant concepts as needed. This talk is based on a recent preprint (https://arxiv.org/abs/2312.02883).