A New Approach to Quantising Space-Time: II. Quantising on a Category of Sets

In a recent paper by the author, a new approach was suggested for quantising space-time, or space. This involved developing a procedure for quantising a system whose configuration space--or history-theory analogue--is the set of objects in a (small) category $\Q$. In the present paper, we show how this theory can be applied to the special case when $\Q$ is a category of sets. This includes the physically important examples where $\Q$ is a category of causal sets, or topological spaces.