Memoirs of the American Mathematical Society 2008; 184 pp; softcover Volume: 193 ISBN10: 0821841424 ISBN13: 9780821841426 List Price: US$80 Individual Members: US$48 Institutional Members: US$64 Order Code: MEMO/193/905
 The primary purpose of this work is to characterise strict \(\omega\)categories as simplicial sets with structure. The author proves the StreetRoberts 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 2categories to this context. Under Street's \(\omega\)categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others. Table of Contents  Simplicial operators and simplicial sets
 A little categorical background
 Double categories, 2categories and \(n\)categories
 An introduction to the decalage construction
 Stratifications and filterings of simplicial sets
 Precomplicial sets
 Complicial sets
 The path category construction
 Complicial decalage constructions
 Street's \(\omega\)categorical nerve construction
 Bibliography
