Abstract
Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of processive plane graphs. We demonstrate the equivalence of Joyal and Street’s graphical calculus and the theory of upward planar drawings. We introduce the category of semi-tensor schemes, and give a construction of a free monoidal category on a semi-tensor scheme. We deduce the unit convention as a kind of quotient construction, and show an idea to generalize the unit convention. Finally, we clarify the relation of the unit convention and Joyal and Street’s construction of a free monoidal category on a tensor scheme.
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Acknowledgements
Xuexing Lu would like to thank Ross Street for comments and encouragement and for John Power for encouragement and a lot of concrete advice for improving this paper. He also would like to thank Maurizio Patrignani for careful check of the proofs in an earlier version of this paper. He also thanks Jinsong Wu for reminding him to change the terminologies and pointing out some mistakes in an earlier version of this paper. This work is partly supported by the National Scientific Foundation of China Nos. 11431010 and 11571329 and “the Fundamental Research Funds for the Central Universities”.
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Lu, X., Ye, Y. & Hu, S. A Graphical Calculus for Semi-Groupal Categories. Appl Categor Struct 27, 163–197 (2019). https://doi.org/10.1007/s10485-018-9549-8
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DOI: https://doi.org/10.1007/s10485-018-9549-8