# Dagger Higher Categories

## Zoomland, 12–15 June 2023, 9:00-13:00 EDT / 15:00-19:00 CEST

Dagger categories, formalized in the early 2000s, describe categories in which every morphism *f*:*X*→*Y* has an adjoint *f*^{†}:*Y*→*X*. They are particularly important for axiomatizing notions like "Hilbert space" and "unitary map", and as such appear vitally in categorical formulations of quantum theory. Quantum theory also requires higher categories, and over the years various groups of people have tried to work out a good definition of "higher dagger categories". The goal of this meeting is to compare these partial definitions, and hopefully to emerge with a complete consensus definition.

This will be an small, invitation-only meeting of experts. The focus is on collaborative conversation. We will have two short (45-minute) talks each day, with two hours of discussion in between. Speakers (not to mention the rest of the audience) will only learn that they are expected to speak a day or two before they talk.

### Confirmed participants

- Bruce Bartlett, Stellenbosch University
- Gio Ferrer, The Ohio State University
- Andre Henriques, University of Oxford
- Chris Heunen, University of Edinburgh
- Brett Hungar, The Ohio State University
- Theo Johnson-Freyd, Dalhousie University and Perimeter Institute
- Cameron Krulewski, Massachusetts Institute of Technology
- Lukas Mueller, Perimeter Institute for Theoretical Physics
- Nivedita Nivedita, University of Oxford
- Dave Penneys, The Ohio State University
- David Reutter, University of Hamburg
- Claudia Scheimbauer, Technical University of Munich
- Peter Selinger, Dalhousie University
- Luuk Stehouwer, Max Planck Institute for Mathematics
- Dominic Verdon, University of Bristol
- Chetan Vuppulury, Sapienza University of Rome

### Schedule

**Monday**

- Luuk Stehouwer: Dagger categories via anti-involutions, and positivity
- Dave Penneys: Manifestly positive delooping

**Tuesday**

- Brett Hungar: Unitary condensation
- Lukas Mueller: (Extended) Reflection Positivity & (Higher) Dagger Categories

**Wednesday**

- Theo Johnson-Freyd: Aut(Cat
_{n}), after Barwick–Schommer-Pries - Gio Ferrer: Operator (i.e. C* or W*)-algebraic tricategories and Gray-categories

**Thursday**

- Dominic Verdun: Higher C*-categories in quantum theory
- David Reutter: Dagger n-categories, a definition