Abstract Tensor Systems as Monoidal Categories

TL;DR

This paper provides a formal categorical framework for Penrose's tensor notation using traced symmetric monoidal categories, establishing a foundational link and simplifying the understanding of their diagrammatic language.

Contribution

It introduces a typed, sum-free abstract tensor system and proves its associated category is the free traced symmetric monoidal category, offering a new formal foundation.

Findings

Constructs a typed, sum-free abstract tensor system

Shows the associated category is the free traced symmetric monoidal category

Provides a simple proof of soundness and completeness for the diagrammatic language

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.