Quotient Categories and Phases

TL;DR

This paper investigates the structure of categories after quotienting by isomorphisms, introducing phased coproducts, with applications in projective geometry and quantum theory, especially in describing superpositions up to phase.

Contribution

It introduces phased coproducts as a weaker notion of coproducts in quotient categories and characterizes categories with these structures as quotients of categories with coproducts.

Findings

Categories with phased coproducts arise as quotients of categories with coproducts.

Application to quantum theory: describes superpositions up to global phase.

Generalizes categorical isotropy in the context of quotient categories.

Abstract

We study properties of a category after quotienting out a suitable chosen group of isomorphisms on each object. Coproducts in the original category are described in its quotient by our new weaker notion of a 'phased coproduct'. We examine these and show that any suitable category with them arises as such a quotient of a category with coproducts. Motivation comes from projective geometry, and also quantum theory where they describe superpositions in the category of Hilbert spaces and continuous linear maps up to global phase. The quotients we consider also generalise those induced by categorical isotropy in the sense of Funk et al.