  We review the problem of finding a general framework within which one can
construct quantum theories of non-standard models for space, or space-time. The
starting point is the observation that entities of this type can typically be
regarded as objects in a category whose arrows are structure-preserving maps.
This motivates investigating the general problem of quantising a system whose
`configuration space' (or history-theory analogue) is the set of objects
$\Ob\Q$ in a category $\Q$.
  We develop a scheme based on constructing an analogue of the group that is
used in the canonical quantisation of a system whose configuration space is a
manifold $Q\simeq G/H$, where $G$ and $H$ are Lie groups. In particular, we
choose as the analogue of $G$ the monoid of `arrow fields' on $\Q$. Physically,
this means that an arrow between two objects in the category is viewed as an
analogue of momentum. After finding the `category quantisation monoid', we show
how suitable representations can be constructed using a bundle (or, more
precisely, presheaf) of Hilbert spaces over $\Ob\Q$. For the example of a
category of finite sets, we construct an explicit representation structure of
this type.
