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

Abstract

This article introduces Hilbert $*$-categories: an abstraction of categories having algebraic and analytic properties similar to those of the categories of real, complex, and quaternionic Hilbert spaces and bounded linear maps. Other examples include categories of Hilbert W*-modules and of unitary representations of groupoids. 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.