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"A survey of graphical languages for monoidal categories", Peter Selinger, 2023

Reviewed October 15, 2023

Citation: Selinger, Peter. "A survey of graphical languages for monoidal categories." New structures for physics (2011): 289-355.

Web: https://arxiv.org/abs/0908.3347

Tags: Mathematical, Expository, Monoidal-categories

This paper gives a nice review of graphical langues for monoidal categories. Every monoidal category has string diagrams associated to it, where strands correspond to objects, boxes correspond to morphisms, and putting boxes next to each other corresponds to the tensor product. Special structures on the category can be given special symbols, e.g. braiding and fusion, which give the graphical language more flavor.

In particular, long and sometimes seemingly arbitrary coherence axioms can be summarized as saying "two diagrams are the same if they are equal up to some form of isotopy". This means that coherence is baked into the notation, which is exactly what you want.

This paper is very nice if you want to get your head around ribbon fusion categories. It works up to them definition by definition, adding more and more structures to your category, and showing at every step why they are necessary. Note that some of the structures introduced are horizontal, and are not needed/automatically implied by the ribbon fusion category axioms.