Dagger categories via anti-involutions and positivity

Abstract: Dagger categories are an essential tool for categorical descriptions of quantum physics, but pose a challenge to category theorists as their definition is in tension with the principle of equivalence'' that lies at the heart of category theory. In this note we propose the alternative, coherent definition of aninvolutive category with a notion of positivity'' and show that the $2$-category formed by these is biequivalent to the $2$-category of dagger categories.