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Categories for the practising physicist | | Bob Coecke
; Eric Oliver Paquette
; | | Date: |
19 May 2009 | | 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. These
are particularly relevant for quantum foundations and for quantum informatics.
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. | | Source: | arXiv, 0905.3010 | | Services: | Forum | Review | PDF | Favorites | |
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