Preface ........................................................ ix
Historical Background .......................................... ix
Relationships to Other Work ................................... xii
Overview and Structure ....................................... xiii
Chapter 1. Simplicial Operators and Simplicial Sets ............. 1
1. Simplicial Operators ......................................... 1
2. The Algebraist's Δ and 2-Categories .......................... 3
3. The Algebraist's Δ and Monoidal Categories ................... 6
4. Simplicial Sets ............................................. 10
5. Semi-Simplicial Sets ........................................ 14
6. Analysing Products of Simplicial Sets - the Theory of
Shuffles .................................................... 15
Chapter 2. A Little Categorical Background ..................... 21
1. Reflective Full Subcategories ............................... 21
2. LFP-Categories and LE-Theories .............................. 25
Chapter 3. Double Categories, 2-Categories and n-Categories .... 33
1. Categories in the Small ..................................... 33
2. Double Categories ........................................... 36
3. 2-Categories and Double Categories with Connections ......... 38
4. n-Categories and ω-Categories ............................... 43
Chapter 4. An Introduction to the Decalage Construction ........ 47
1. Nerves and Decalage ......................................... 47
2. Comonad Transformations and Simplicial Reconstruction ....... 49
Chapter 5. Stratifications and Filterings of Simplicial Sets ... 55
1. Stratified Simplicial Sets .................................. 55
2. Superstructures and Filtered Semi-Simplicial Sets ........... 61
Chapter 6. Pre-Complicial Sets ................................. 65
1. Introducing Pre-Complicial Sets ............................. 65
2. Tensor Products of Pre-Complicial Sets ...................... 66
3. Pre-Tensors and Preservation of t-Extensions ................ 71
4. Some Other Preservation Properties .......................... 78
5. A Monoidal Biclosed Structure on Pre-Complicial sets ........ 80
6. Superstructures of Pre-Complicial Sets ...................... 83
Chapter 7. Complicial Sets ..................................... 85
1. Introducing Complicial Sets ................................. 85
2. Glueing Squares and Filling Lemmas .......................... 86
3. Tensor Products and Complicial Sets ......................... 90
4. Superstructures of Complicial Sets ......................... 102
Chapter 8. The Path Category Construction ..................... 105
1. The Complicial Category of Prisms .......................... 105
2. Path Categories and Superstructures ........................ 108
3. A Complicial Double Category with Connections .............. 110
Chapter 9. Complicial Decalage Constructions .................. 115
1. A Decalage Construction on Complicial Sets ................. 115
2. A Path Construction on Complicially Enriched Categories .... 120
3. A Decalage Construction on Complicially Enriched
Categories ................................................. 125
4. Semi-Simplicial Reconstruction ............................. 127
Chapter 10. Street's ω-Categorical Nerve Construction ......... 133
1. Parity Complexes ........................................... 133
2. Collapsers and Stratified Parity Complexes ................. 147
3. ω-Categorical Nerve Constructions .......................... 150
4. Products of Parity Complexes and the Complicial Tensor ..... 155
5. An Inductive Proof of the Street-Roberts Conjecture ........ 163
Bibliography .................................................. 173
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