An embedding theorem for Hilbert categories

TL;DR

This paper introduces axioms for pre-Hilbert categories and proves they can be embedded into categories of pre-Hilbert spaces, extending to Hilbert spaces under certain conditions, without assuming complex fields or enrichment.

Contribution

It provides a new axiomatic framework for (pre-)Hilbert categories and proves an embedding theorem into categories of (pre-)Hilbert spaces, broadening the understanding of categorical Hilbert space structures.

Findings

Pre-Hilbert categories embed into pre-Hilbert spaces preserving structure.

Scalars in such categories form an involutive field.

Embedding extends to Hilbert spaces with continuous linear maps.

Abstract

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all finite (co)limits. An intermediate result that is important in its own right is that the scalars in such a category necessarily form an involutive field. In case of a Hilbert category, the embedding extends to the category of Hilbert spaces and continuous linear maps. The axioms for (pre-)Hilbert categories are weaker than the axioms found in other approaches to axiomatizing 2-Hilbert spaces. Neither enrichment nor a complex base field is presupposed. A comparison to other approaches will be made in the introduction.