Dagger categories via anti-involutions and positivity

Abstract

Dagger categories are an essential tool for categorical descriptions of quantum physics, for example in categorical quantum mechanics and unitary topological field theory. Their definition however is in tension with the ``principle of equivalence'' that lies at the heart of category theory, thereby inhibiting generalizations to higher categories. In this note we propose an alternative, coherent description of dagger categories based on the well-studied notion of anti-involutions $d\colon \mathcal{C} \to \mathcal{C}^{op}$, which coherently square to the identity functor $\eta\colon d^2 \cong \operatorname{id}_{\mathcal{C}}$. A general anti-involution need not be the identity on objects, but we instead consider certain isomorphisms $dx \cong x$, which we call Hermitian fixed points as they generalize the notion of a Hermitian inner product on a vector space. We define a ``positivity notion" on $(\mathcal{C},d, \eta)$ in terms of such Hermitian fixed points. This terminology is motivated by the dagger category of Hilbert spaces, in which case the positivity notion consists of the positive definite pairings. Our main result is that the $2$-category of anti-involutive categories with a positivity notion is biequivalent to the $2$-category of dagger categories.