Categories for the Practising Physicist
Categories for the Practising Physicist
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DOI：
10.1007/978-3-642-12821-9_3
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发表时间：
2011-01-01
期刊：
影响因子：
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通讯作者：
Paquette, E. O.
中科院分区：
文献类型：
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作者：
Coecke, B.;Paquette, E. O.
在本章中，我们以一种非常规的方式考察了范畴论中的一些特定主题。我们的主要关注点是幺半群类别，主要是对称类别，我们为此提出了物理解释。特别关注以有限维希尔伯特空间为对象、以线性映射为态射、以张量积为其幺群结构 (FdHilb) 的范畴。我们还详细讨论了以集合作为对象、以关系作为态射、以笛卡尔积作为其幺半群结构 (Rel) 的范畴，以及第三，以流形作为对象以及它们之间的共边作为态射 (2Cob) 的范畴。虽然集合、希尔伯特空间和流形不具有任何非平凡的共同结构，但这三个类别实际上在结构上非常相似。共同的特征是图解演算、紧凑的封闭结构和特定种类的内部共基函数，这些特征在它们中都发挥着重要作用。此外，类别 FdHilb 和 Rel 允许分类矩阵演算。这些特征共同引导我们走向拓扑量子场论。我们还讨论了位位类别、组表征实际上是类别构造，以及幺半群类别的严格性和连贯性是什么。在我们试图补充现有文献的过程中，我们省略了一些非常基本的主题。对于这些，我们建议读者参考其他可用的资源。
In this chapter we survey some particular topics in category theory in a somewhat unconventional manner. Our main focus will be on monoidal categories, mostly symmetric ones, for which we propose a physical interpretation. Special attention is given to the category which has finite dimensional Hilbert spaces as objects, linear maps as morphisms, and the tensor product as its monoidal structure (FdHilb). We also provide a detailed discussion of the category which has sets as objects, relations as morphisms, and the cartesian product as its monoidal structure (Rel), and thirdly, categories with manifolds as objects and cobordisms between these as morphisms (2Cob). While sets, Hilbert spaces and manifolds do not share any non-trivial common structure, these three categories are in fact structurally very similar. Shared features are diagrammatic calculus, compact closed structure and particular kinds of internal comonoids which play an important role in each of them. The categories FdHilb and Rel moreover admit a categorical matrix calculus. Together these features guide us towards topological quantum field theories. We also discuss posetal categories, how group representations are in fact categorical constructs, and what strictification and coherence of monoidal categories is all about. In our attempt to complement the existing literature we omitted some very basic topics. For these we refer the reader to other available sources.