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Complicial Sets Characterising the Simplicial Nerves of Strict $\omega$-Categories

Dominic Verity Macquarie University, Sydney, Australia
Available Formats:
Electronic ISBN: 978-1-4704-0511-3
Product Code: MEMO/193/905.E
List Price: $80.00 MAA Member Price:$72.00
AMS Member Price: $48.00 Click above image for expanded view Complicial Sets Characterising the Simplicial Nerves of Strict$\omega$-Categories Dominic Verity Macquarie University, Sydney, Australia Available Formats:  Electronic ISBN: 978-1-4704-0511-3 Product Code: MEMO/193/905.E  List Price:$80.00 MAA Member Price: $72.00 AMS Member Price:$48.00
• Book Details

Memoirs of the American Mathematical Society
Volume: 1932008; 184 pp
MSC: Primary 18; 55;

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.

• Chapters
• 1. Simplicial operators and simplicial sets
• 2. A little categorical background
• 3. Double categories, 2-categories and $n$-categories
• 4. An introduction to the decalage construction
• 5. Stratifications and filterings of simplicial sets
• 6. Pre-complicial sets
• 7. Complicial sets
• 8. The path category construction
• 9. Complicial decalage constructions
• 10. Street’s $\omega$-categorical nerve construction
• Requests

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Volume: 1932008; 184 pp
MSC: Primary 18; 55;

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.

• Chapters
• 1. Simplicial operators and simplicial sets
• 2. A little categorical background
• 3. Double categories, 2-categories and $n$-categories
• 4. An introduction to the decalage construction
• 5. Stratifications and filterings of simplicial sets
• 6. Pre-complicial sets
• 7. Complicial sets
• 8. The path category construction
• 9. Complicial decalage constructions
• 10. Street’s $\omega$-categorical nerve construction
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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