## Higher-dimensional Mac Lane's pentagon and Zamolodchikov equations (1999)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Crans99higher-dimensionalmac,

author = {Sjoerd E. Crans},

title = {Higher-dimensional Mac Lane's pentagon and Zamolodchikov equations},

institution = {},

year = {1999}

}

### OpenURL

### 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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Citation Context ...ur Yang-Baxter hexagons. Hence these 3-arrows can be interpreted as proofs of Zamolodchikov equations in (somewhat weak) braided 2-dimensional teisi. Comparing this with Kapranov and Voevodsky's work =-=[10, 9, 1-=-1], whose braided monoidal 2-categories dier slightly from braided 2-dimensional teisi, the conclusion will be that there are not eight, not three, butsve dierent equations involving the 2-arrows S + ... |

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Citation Context ...he support of NSERC and FCAR Quebec 1 Before any of this can be generalized to higher dimensions, one needs a good candidate for what weak n-categories might strictify to. One such candidate is teisi =-=[5]-=-. In a tas, m-composition of a p-arrow and a q-arrow for p; q > m+ 1 results in a (p + q m 1)-arrow, which generalizes the 0-composition of two 2-arrows in a Gray-category resulting in a 3-arrow, as i... |

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