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Categories for Quantum TheoryAn Introduction$
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Chris Heunen and Jamie Vicary

Print publication date: 2019

Print ISBN-13: 9780198739623

Published to Oxford Scholarship Online: January 2020

DOI: 10.1093/oso/9780198739623.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 17 April 2021

Monoidal 2-Categories

Monoidal 2-Categories

(p.265) 8 Monoidal 2-Categories
Categories for Quantum Theory

Chris Heunen

Jamie Vicary

Oxford University Press

Monoidal 2-categories are higher-dimensional versions of monoidal categories, allowing a more expressive syntax that plays an important role in modern mathematics. We explore their two-dimensional graphical calculus, and show how duality gives a language for oriented surfaces, from which Frobenius algebras emerge in a natural way. We describe 2-Hilbert spaces, categorifications of Hilbert spaces and explore the monoidal 2-category 2Hilb that they give rise to. We then show how we can use dualities in 2Hilb to give a concise and purely topological language to reason about teleportation, dense coding and complementarity.

Keywords:   Monoidal 2-category, Categorification, Oriented duality, 2-Hilbert space, Deligne tensor product

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