Complicial sets characterising the simplicial nerves of strict ω-categories

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The primary purpose of this work is to characterise strict ω-categories as simplicial sets with structure. The author proves the Street-Roberts conjecture in the form formulated by Ross Street in his work on Orientals, which states that they are exactly the "complicial sets" defined and named by John Roberts in his handwritten notes of that title (circa 1978). On the way the author substantially develops Roberts' theory of complicial sets itself and makes contributions to Street's theory of parity complexes. In particular, he studies a new monoidal closed structure on the category of complicial sets which he shows to be the appropriate generalisation of the (lax) Gray tensor product of 2-categories to this context. Under Street's ω-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.
Original languageEnglish
Place of PublicationProvidence, RI
PublisherAmerican Mathematical Society
ISBN (Print)9780821841426
Publication statusPublished - 2008

Publication series

NameMemoirs of the American Mathematical Society
PublisherAmerican Mathematical Society
ISSN (Print)0065-9266


  • Categories (Mathematics)
  • Algebraic topology


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