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arXiv:1308.3586math.CT

Abstract Tensor Systems as Monoidal Categories

Aleks Kissinger

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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.

arXiv comments
Dedicated to Joachim Lambek on the occasion of his 90th birthday
Journal reference
DOI
Subjects
math.CTCategory Theory math-phMathematical Physics

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  1. Einsum for Tensor Manipulation 1 comment

    • Working Google Scholar link: https://scholar.google.com/scholar?hl=en&as_sdt=7%2C39&q=tra... Direct arXiv PDF link to first result: https://arxiv.org/pdf/1308.3586

      jasomillView on HNParent context

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