Quantising on a Category

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 ≃ 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.