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

Abstract

This article introduces Hilbert $\ast$-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 $\ast$-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.

Added proof that Hilbert *-categories are Douglian, and thus every contraction in a Hilbert *-category has a codilator