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Abstract Tensor Systems as Monoidal Categories

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Categories and Types in Logic, Language, and Physics

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8222))

Abstract

The primary contribution of this paper is to give a formal, categorical treatment to Penrose’s abstract tensor notation, in the context of traced symmetric monoidal categories. To do so, we introduce a typed, sum-free version of an abstract tensor system and demonstrate the construction of its associated category. We then show that the associated category of the free abstract tensor system is in fact the free traced symmetric monoidal category on a monoidal signature. A notable consequence of this result is a simple proof for the soundness and completeness of the diagrammatic language for traced symmetric monoidal categories.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Oxford, UK

    Aleks Kissinger

Authors
  1. Aleks Kissinger
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Editor information

Editors and Affiliations

  1. Department of Philosophy, Education and Quantitative Economics, Gabriele D’Annunzio University, Chieti-Pescara, Via dei Vestini 31, 60100, Chieti, Italy

    Claudia Casadio

  2. Department of Computer Science, Oxford University, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Bob Coecke

  3. Utrecht Institute of Linguistics OTS, Utrecht University, Trans 10,, 3512 JK, Utrecht, The Netherlands

    Michael Moortgat

  4. Department of Mathematics, University of Ottawa, 585 King Edward, K1N 6N5, Ottawa, ON, Canada

    Philip Scott

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Kissinger, A. (2014). Abstract Tensor Systems as Monoidal Categories. In: Casadio, C., Coecke, B., Moortgat, M., Scott, P. (eds) Categories and Types in Logic, Language, and Physics. Lecture Notes in Computer Science, vol 8222. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54789-8_13

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  • DOI: https://doi.org/10.1007/978-3-642-54789-8_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-54788-1

  • Online ISBN: 978-3-642-54789-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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