  The problem of extending the insights and techniques of categorical quantum
mechanics to infinite-dimensional systems was considered in (Coecke and Heunen,
2016). In that work the $\mathrm{CP}^{\infty}$-construction, which recovers the
category of Hilbert spaces and quantum operations from the category of Hilbert
spaces and bounded linear maps, was defined. Here we show that by a `horizontal
categorification' of the $\mathrm{CP}^{\infty}$-construction, one can recover
the category of all von Neumann algebras and channels (normal unital completely
positive maps) from the 2-category $[W^*]$ of von Neumann algebras, bimodules
and intertwiners. As an application, we extend Choi's characterisation of
extremal channels between finite-dimensional matrix algebras to a
characterisation of extremal channels between arbitrary von Neumann algebras.
