A graphical calculus for semi-groupal categories

TL;DR

This paper develops a graphical calculus for semi-groupal categories, extending Joyal and Street's work on monoidal categories, and introduces frameworks and constructions to formalize and relate these graphical representations.

Contribution

It introduces topological and combinatorial frameworks for semi-groupal graphical calculus, and connects these to existing theories of monoidal categories and upward planar drawings.

Findings

Established equivalence between graphical calculus and upward planar drawings.

Constructed the category of semi-tensor schemes and free monoidal categories.

Linked the unit convention to quotient constructions and existing monoidal category theories.

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.