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

This talk will introduce Hilbert $\ast$-categories [1, 2]—an abstraction capturing algebraic and analytic aspects of the categories ${\mathrm{𝐇𝐢𝐥𝐛}}_{ℝ}$, ${\mathrm{𝐇𝐢𝐥𝐛}}_{ℂ}$ and ${\mathrm{𝐇𝐢𝐥𝐛}}_{ℍ}$ of real, complex and quaternionic Hilbert spaces and bounded linear maps. Additional examples include:

• •

  for each von Neumann algebra $A$, the category ${\mathrm{𝐇𝐢𝐥𝐛}}_A$ of self-dual Hilbert $A$-modules; and,

• •

  for each group $G$, the category ${\mathrm{𝐔𝐑𝐞𝐩}}_G$ of unitary representations of $G$.

Hilbert $\ast$-categories are “analytically” complete in two ways:

1. (i)

   every bounded increasing net of Hermitian endomorphisms has a supremum, and

2. (ii)

   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:

1. (i)

   codirected ${\mathrm{\ell }}^2$-limits of contractions, and

2. (ii)

   ${\mathrm{\ell }}^2$-products.

In turn, these universal constructions are built from directed colimits in the wide subcategory of isometries.