Bonn Topology Group - Abstracts

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Talk

January 7, 2025
Luuk Stehouwer (Dalhousie University): Higher dagger categories

Abstract

Consider the category of finitely generated projective modules over a commutative ring equipped with a nondegenerate inner product. The transpose of a linear map makes this into a dagger category, i.e. a category C equipped with a functor \dagger: C \to C^op which is the identity on objects and \dagger^2 = id_C strictly. In modern homotopy theory, it is desirable to understand higher-algebraic analogues of R-modules with Hermitian forms. The oriented bordism category is another important example of a dagger category for which a generalization to (\infty,n)-categories is interesting for topologists. At first sight it seems to be difficult to categorify the very strict notion of being an identity-on-objects involution. In joint work with a large group of people, we provided a definition of dagger (\infty,n)-categories. In this talk I will sketch this definition and show how the bordism (\infty,n)-category with a stable tangential structure is a symmetric monoidal (\infty,n)-dagger category. If time permits, I will explain work in progress generalizing the cobordism hypothesis to the setting of dagger categories.

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