Bennett and Stinespring, Together at Last

We present a universal construction that relates reversible dynamics on open systems to arbitrary dynamics on closed systems: the well-pointed restriction affine completion of a monoidal restriction category. This categorical completion encompasses both quantum channels, via Stinespring dilation, and classical computing, via Bennett's method. Moreover, in these two cases, we show how our construction can be 'undone' by a further universal construction. This shows how both mixed quantum theory and classical computation rest on entirely reversible foundations.