Higher-dimensional Mac Lane's pentagon and Zamolodchikov equations

AbstractAn important ingredient of Mac Lane's coherence theorem for monoidal categories is Mac Lane's pentagon, a diagram whose commutativity is needed so that “all diagrams commute”. This paper gives a higher-dimensional generalization of Mac Lane's pentagon: a 6-dimensional diagram whose commutativity is needed in order for all diagrams in somewhat weak teisi to commute.Looping twice gives a 4-dimensional diagram in somewhat weak braided teisi, of which five 3-dimensional edges can be interpreted as proofs of five different Zamolodchikov equations in braided monoidal 2-categories. Hence higher-dimensional Mac Lane's pentagon expresses the relations between these proofs concisely