Contents
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 A and 2-Categories 3
3. The Algebraist's A 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 ^-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
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