A Graphical Calculus for Semi-Groupal Categories

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.

Graded Medial n-Ary Algebras and Polyadic Tensor Categories

“…We call a category C a strict (binary) semigroupal [52,53] (or strictly associative semigroupal category [54], also, semi-monoidal [55]), if the bifunctor M p2bq satisfies only (without unit objects and unitors) the binary associativity condition pX…”

“…In the case of a non-strict semigroupal category SGCat (with no unit objects and unitors) [52,54] (see, also, [53,58,59]) a collection of mappings can be introduced which are just the isomorphisms (associators) A p3bq "…”

“…However, if the associator A p3bq satisfies some consistency relations, they can give isomorphic results, such that the corresponding diagrams commute, which is the statement of the coherence theorem [42,60]. This can also be applied to SGCat, because it can be proved independently of existence of units [52][53][54]. It was shown [42] that it is sufficient to consider one commutative diagram using the associator (the associativity constraint) for two different rearrangements of parentheses for three tensor multiplications of four objects, giving the following isomorphism…”

“…We call sequences of objects and morphisms X-polyads and f-polyads [16] and denote them X and f, respectively (as in (53)).…”