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Complicial Sets Characterising the Simplicial Nerves of Strict $$\omega$$-Categories
Dominic Verity, Macquarie University, Sydney, Australia
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Memoirs of the American Mathematical Society
2008; 184 pp; softcover
Volume: 193
ISBN-10: 0-8218-4142-4
ISBN-13: 978-0-8218-4142-6
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 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 $$\omega$$-categorical nerve construction, which the author shows to be an equivalence, this tensor product coincides with those of Steiner, Crans and others.

• Simplicial operators and simplicial sets
• A little categorical background
• Double categories, 2-categories and $$n$$-categories
• An introduction to the decalage construction
• Stratifications and filterings of simplicial sets
• Pre-complicial sets
• Complicial sets
• The path category construction
• Complicial decalage constructions
• Street's $$\omega$$-categorical nerve construction
• Bibliography