Abstract

An 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 ve 3-dimensional edges can be interpreted as proofs of ve dierent Zamolodchikov equations in braided monoidal 2-categories. Hence higher-dimensional Mac Lane's pentagon expresses the relations between these proofs concisely. 1 Introduction The coherence theorem for tricategories states that every tricategory is triequivalent to a Gray-category [6]. But there is also another coherence theorem for tricategories, stating that tricategories are (algebras for a) contractible (operad) [1], which roughly says that \all diagrams in a tricategory...

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