Abstract: I will introduce the homotopy-theoretic classification of fully extended reflection-positive invertible field theories and their application to classifying symmetry-protected topological phases (SPTs) in physics. In ongoing work with Arun Debray, Natalia Pacheco-Tallaj, and Luuk Stehouwer, we use twisted index maps to relate invertible field theories and K-theory, modeling a physics process comparing free and interacting SPTs. In our example of interest, we witness the "Morita variance" of invertible field theories, in which Bott periodicity in K-theory is mapped to a non-periodic "Bott spiral" of invertible field theories.

Markov categories are a surprisingly powerful categorical framework for probability theory and statistics. But in order to be taken seriously, any such framework needs to provide an account of the fundamental theorems of probability theory: the law of large numbers. In this talk, I will give an introduction to Markov categories and explain how to formulate a categorical version of the law of large numbers. This revolves around the idea of empirical sampling, by which we mean drawing a uniformly random element from an infinite sequence of samples. I will explain how to make this idea precise in categorical terms, what it has to do with randomly permuting an infinite sequence, and how to formulate a strong law of large numbers (Glivenko–Cantelli theorem) in terms of it. There are close connections with restriction categories and with de Finetti's theorem.

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Abstract: TBA

I shall discuss structures which arise in the definition of biprops – weakly category enriched coloured props. In particular, the set of operations (Xi)i∈I → (Yj)j∈J (Xi and Yj are colours) is a category, and the composition of operations is a functor. We relate the notion of a biprop with the notion of a weak Cat-enriched symmetric multicategory. Namely, for a given weak Cat-enriched symmetric multicategory we construct a biprop so that to a multifunctor of such multicategories there corresponds a morphism of biprops.

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