Pair production makes sense with unbracketed string objects, as in the basic particle diagrams above the green lines. To obtain the (color) Jacobi rule, however, we need bracketed objects. Recall that Bar-Natan replaced the bracket trees by grouped strands in a category of nonassociative braids. Usually in a (braided) monoidal category, parallel object strings are specified as tensor products, as shown on the left below. The actual multiplication map $M: A \otimes A \rightarrow A$ happens outside the diagrams under consideration.

But for an algebra object $A$, we might like to represent the multiplication map $m$ as a binary tree node. This alters the representation of the associator map, to the new form on the right below. Note that it requires the existence of a map $A \rightarrow A \otimes A$, but that is OK if all our algebras are Hopf algebras.

Now strand groupings in a braid morphism represent both multiplications (resp. comultiplications) and bracketings (resp. unbracketings), meaning that the two processes occur together. The distinction between a word of length $3$ in $A$ and the object $A \otimes A \otimes A$ is given by the distance between strands, much as Bar-Natan envisaged.

This idea may be horrifying at first, since algebra objects usually live in the world of string diagrams, whereas bracketing nodes live in the dual world of arrows. However, we are particularly interested in categories with multiple duality structures, such as that between strings and arrows. Physical categories will by default be high dimensional, with many levels of arrows, so we need a number of shortcuts to represent different composition types.

6 years ago

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