Unitary, Inner product, and Dagger categories

TL;DR

This paper characterizes dagger categories via inner product categories, showing their equivalence to unitary categories and exploring metric-preserving maps within this framework.

Contribution

It provides an alternative characterization of dagger categories using inner product categories and establishes their 2-categorical equivalence with unitary categories.

Findings

Inner product categories are equivalent to unitary categories.

Unitary categories can be strictified to dagger categories.

The framework allows defining metric-preserving and special maps.

Abstract

This article provides an alternate characterization of dagger categories, which are central to the study of categorical quantum mechanics, in terms of inner product categories. An inner product category is an "achiral involutive" category with an inner product combinator. Inner product categories are, in turn, precisely the same as unitary categories, which are a weaker form of dagger categories. In unitary categories, there is an isomorphism between an object and its dagger, instead of the identity function as in the case of dagger categories. Every unitary category is equipped with a global inner product structure, which allows one to strictify the involutive structure on the unitary category to obtain a dagger category, making unitary categories 2-categorically equivalent to dagger categories. By regarding the inner product as an abstract metric on an (achiral) involutive category, one can define metric-preserving maps (isometries) in inner product categories, and also develop other notions of special maps -- unitary, Hermitian, positive, and normal maps -- in this setting.