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.
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机译：对于量化因果集，拓扑或其他此类时空模型的问题，提出了一种新方法。起点是观察到可以将这种类型的实体视为其箭头是保留结构的映射的类别中的对象。这激发了对量化系统的一般问题的研究，该系统的“配置空间”（或历史理论类似物）可以视为类别中的一组对象。在系列文章的第一篇中，我们通常研究此问题，并基于构造组的类似物来开发一种方案，该组用于配置空间为流形$ Q \ simeq G / H的系统的标准量化$其中G和H是李群。特别是，我们选择类别上“箭头字段”的类人半身像作为G的类似物。从物理上讲，这意味着类别中两个对象之间的箭头被视为某种形式的动量。找到“类别量化半定式”后，我们展示了如何使用一组对象上的希尔伯特空间来构造合适的表示。
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