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Categories for the Practising Physicist

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New Structures for Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 813))

Abstract

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.

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Notes

  1. 1.

    In the light of the previous footnote, note here that this law applies to any reasonable notion of equality for processes.

  2. 2.

    Paper [20] provided a conceptual template for setting up the content of this paper. However, here we go in more detail and provide more examples.

  3. 3.

    Typically, “family” will mean a class rather than a set. While for many constructions the size of \(|\textbf{C}|\) is important, it will not play a key role in this paper.

  4. 4.

    In order to conceive Cat as a concrete category, the family of objects should be restricted to the so-called “small” categories i.e., categories for which the family of objects is a set.

  5. 5.

    The first time the 1st author heard about categories was in a Philosophy of Science course, given by a biologist specialised in population dynamics, who discussed the importance of category theory in the influential work of Robert Rosen [59].

  6. 6.

    Note that this operation on morphisms is a typed variant of the notion of monoid.

  7. 7.

    This \(90^\circ\) rotation is merely a consequence of our convention to read pictures from bottom-to-top. Other authors obey different conventions e.g. top-to-bottom or left-to-right.

  8. 8.

    Naturality is one of the most important concepts of formal category theory. In fact, in the founding paper [33] Eilenberg and MacLane argue that their main motivation for introducing the notion of a category is to introduce the notion of a functor, and that their main motivation for introducing the notion of a functor is to introduce the notion of a natural transformation.

  9. 9.

    There is no particular reason why we ask for biproducts to be specified while in the case of Cartesian categories we only required existence. This is a matter of taste, whether one prefers “being Cartesian” or “being a biproduct category” to be conceived as a “property a category possesses” or “some extra structure it comes with”. There are different “schools” of category theory which have strong arguments for either of these. Each of these have their virtues and therefore we decided to give an example of both.

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Acknowledgments

We very much appreciated the feedback from the n-category cafe on a previous draft of this paper, by John Baez, Hendrik Boom, Dave Clarke, David Corfield and Aaron Lauda. We in particular thank Frank Valckenborgh for proofreading the final version.

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  1. School of Computer Science, McGill University, Montreal, QC, Canada

    É.O. Paquette

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  1. B. Coecke
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  2. É.O. Paquette
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    Bob Coecke

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Coecke, B., Paquette, É. (2010). Categories for the Practising Physicist. In: Coecke, B. (eds) New Structures for Physics. Lecture Notes in Physics, vol 813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12821-9_3

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  • DOI: https://doi.org/10.1007/978-3-642-12821-9_3

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