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

CP$^{\infty}$ and beyond: 2-categorical dilation theory