In [1], 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, Ob(Q), in a (small) category Q. The
quantum states in this approach are cross-sections of a bundle A
is in K[A] of Hilbert spaces over Ob(Q).
The Hilbert spaces K[A], A are in
Ob(Q)], depend strongly on
the object A, and have to be chosen so as to get an irreducible,
faithful, representation of the basic `category quantisation monoid'.
In the present paper, we develop a different approach in which the
state vectors are complex-valued functions on the set of arrows in Q.
This throws a new light on the Hilbert bundle scheme: in particular,
we recover the results of that approach in the, physically important,
example when Q is a small category of finite sets.