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The Graphical Language of Symmetric Traced Monoidal Categories

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Traditionally, diagrams have not been treated as first-class citizens in mathematics. However, more recently a plethora of diagrammatic languages have been introduced, for systems such as quantum computation, computational linguistics and signal flow graphs. These graphical languages build on a common infrastructure of (usually symmetric and strict) monoidal categories, and in particular compact closed categories. One class of systems that do not fit the compact closed framework are those with a clearly defined notion of input and output, such as digital circuits. These systems require a different kind of categorical setting, namely that of a symmetric traced monoidal category (STMC).

We therefore define a graphical language for STM Cs, using a variant on hypergraphs that we call ‘linear hypergraphs’. This language is sound and complete – any morphism in the STMC can be interpreted as a well-formed linear hypergraph up to isomorphism, and any linear hypergraph is the representation of a unique morphism, up to the equational theory of the category.

We can then express the axioms of our monoidal theory as graph rewrite rules – we show how we can use our graphical language to apply the framework of double pushout (DPO) rewriting to act as a graph rewriting diagrammatic semantics.

This talk is part of the Bravo series.

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