Hilbert $*$-categories: Where limits in analysis and category theory meet

TL;DR

This paper introduces Hilbert $*$-categories, an abstract framework capturing algebraic and analytic properties of Hilbert spaces and modules, and establishes their completeness via new universal constructions.

Contribution

It defines Hilbert $*$-categories and shows their analytic completeness results are derived from novel universal constructions involving $ ext{ell}^2$-limits and products.

Findings

Hilbert $*$-categories generalize Hilbert space categories.

Completeness properties are derived from $ ext{ell}^2$-limits and products.

Examples include categories of Hilbert W*-modules and unitary representations.

Abstract

This article introduces Hilbert $*$-categories: an abstraction of categories with similar algebraic and analytic properties to the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include categories of Hilbert W*-modules and of unitary group-representations. Hilbert $*$-categories are "analytically" complete in two ways: every bounded increasing sequence of Hermitian endomorphisms has a supremum, and every suitably bounded orthogonal family of parallel morphisms is summable. These "analytic" completeness properties are not assumed outright; rather, they are derived, respectively, from two new universal constructions: codirected $\ell^2$-limits of contractions and $\ell^2$-products. In turn, these are built from directed colimits in the wide subcategory of isometries.