  A new approach is suggested to the problem of quantising causal sets, or
topologies, or other such models for space-time (or space). The starting point
is the observation that entities of this type can 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) can be regarded as the set of objects in a
category. In this first of a series of papers, we study this question in
general and 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 the category.
Physically, this means that an arrow between two objects in the category is
viewed as some sort of analogue of momentum. After finding the `category
quantisation monoid', we show how suitable representations can be constructed
using a bundle of Hilbert spaces over the set of objects.
