Dagger $n$-categories
Giovanni Ferrer, Brett Hungar, Theo Johnson-Freyd, Cameron Krulewski, Lukas Müller, Nivedita, David Penneys, David Reutter, Claudia Scheimbauer, Luuk Stehouwer, Chetan Vuppulury
TL;DR
This work develops a coherent framework for dagger higher categories by defining dagger $(\infty,n)$-categories via flaggings and involutive symmetries, and shows how to realize them concretely in bordism contexts. The authors present multiple equivalent viewpoints—flagged, enriched, and bicategorical—leading to a robust higher-categorical notion of unitary duals and adjoints. A central application is equipping bordism categories with a PL$(n)$-dagger structure to model reflection-positive, unitary extended TQFTs, with potential targets like higher Hilbert-space categories. This framework aims to underpin unitarity in quantum field theory alongside locality via higher category theory, offering a foundation for unitary topological quantum field theories and related structures.
Abstract
Category theory provides a unified language for organizing composable operations in many disciplines. In disciplines where unitarity is fundamental -- such as functional analysis, quantum field theory, and quantum logic -- this language must also capture adjoints, leading to the notion of dagger categories. Higher category theory, which extends this framework to encode operations between operations, has recently become indispensable in both theoretical physics and pure mathematics. Finding a higher categorical analogue of a dagger category is therefore key to the foundations of quantum field theory. In this work, we present a coherent definition of \emph{dagger $(\infty,n)$-category} in terms of equivariance data trivialized on parts of the category. Our main example is the bordism $(\infty,n)$-category $\mathbf{Bord}_{n}^X$. This allows us to define (fully-local) \emph{reflection-positive topological quantum field theories} to be higher dagger functors out of $\mathbf{Bord}_{n}^X$.