  Classical block designs are important combinatorial structures with a wide
range of applications in Computer Science and Statistics. Here we give a new
abstract description of block designs based on the arrow category construction.
We show that models of this structure in the category of matrices and natural
numbers recover the traditional classical combinatorial objects, while models
in the category of completely positive maps yield a new definition of quantum
designs. We show that this generalizes both a previous notion of quantum
designs given by Zauner and the traditional description of block designs.
Furthermore, we demonstrate that there exists a functor which relates every
categorical block design to a quantum one.
