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Dagger $n$-categories (2403.01651v2)

Published 4 Mar 2024 in math.CT, math-ph, math.AT, math.MP, and math.QA

Abstract: We present a coherent definition of dagger $(\infty,n)$-category in terms of equivariance data trivialized on parts of the category. Our main example is the bordism higher category $\mathbf{Bord}{n}X$. This allows us to define a reflection-positive topological quantum field theory to be a higher dagger functor from $\mathbf{Bord}{n}X$ to some target higher dagger category $\mathcal{C}$. Our definitions have a tunable parameter: a group $G$ acting on the $(\infty,1)$-category $\mathbf{Cat}{(\infty,n)}$ of $(\infty,n)$-categories. Different choices for $G$ accommodate different flavours of higher dagger structure; the universal choice is $G = \operatorname{Aut}(\mathbf{Cat}{(\infty,n)}) = (\mathbb{Z}/2\mathbb{Z})n$, which implements dagger involutions on all levels of morphisms. The Stratified Cobordism Hypothesis suggests that there should be a map $\mathrm{PL}(n) \to \operatorname{Aut}(\mathbf{AdjCat}{(\infty,n)})$, where $\mathrm{PL}(n)$ is the group of piecewise-linear automorphisms of $\mathbb{R}n$ and $\mathbf{AdjCat}{(\infty,n)}$ the $(\infty,1)$-category of $(\infty,n)$-categories with all adjoints; we conjecture more strongly that $\operatorname{Aut}(\mathbf{AdjCat}{(\infty,n)}) \cong \mathrm{PL}(n)$. Based on this conjecture we propose a notion of dagger $(\infty,n)$-category with unitary duality or $\mathrm{PL}(n)$-dagger category. We outline how to construct a $\mathrm{PL}(n)$-dagger structure on the fully-extended bordism $(\infty,n)$-category $\mathbf{Bord}_nX$ for any stable tangential structure $X$; our outline restricts to a rigorous construction of a coherent dagger structure on the unextended bordism $(\infty,1)$-category $\mathbf{Bord}{n,n-1}X$. The article is a report on the results of a workshop held in Summer 2023, and is intended as a sketch of the big picture and an invitation for more thorough development.

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