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Simplicial and Dendroidal Homotopy Theory 1st ed. 2022Suurem pilt
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Püsilink: https://www.kriso.ee/db/9783031104466.html
This open access book offers a self-contained introduction to the homotopy theory of simplicial and dendroidal sets and spaces. These are essential for the study of categories, operads, and algebraic structure up to coherent homotopy. The dendroidal theory combines the combinatorics of trees with the theory of Quillen model categories.
 
Dendroidal sets are a natural generalization of simplicial sets from the point of view of operads. In this book, the simplicial approach to higher category theory is generalized to a dendroidal approach to higher operad theory. This dendroidal theory of higher operads is carefully developed in this book. The book also provides an original account of the more established simplicial approach to infinity-categories, which is developed in parallel to the dendroidal theory to emphasize the similarities and differences.
 
Simplicial and Dendroidal Homotopy Theory is a complete introduction, carefully written with the beginning researcher in mind and ideally suited for seminars and courses. It can also be used as a standalone introduction to simplicial homotopy theory and to the theory of infinity-categories, or a standalone introduction to the theory of Quillen model categories and Bousfield localization.

Arvustused

This book is a readable and carefully organized account of dendroidal sets by two of the main figures in the field. It gives a self-contained, detailed description of dendroidal sets and spaces . Each chapter is also accompanied by a short section of historical notes giving background, references, and historical perspectives on the ideas presented. (Ben C Walter, Mathematical Reviews, December, 2023)

Part I The Elementary Theory of Simplicial and Dendroidal Sets
1 Operads
3(46)
1.1 Operads
3(7)
1.2 Algebras for Operads
10(6)
1.2.1 Definitions and Examples
10(4)
1.2.2 Free Algebras
14(1)
1.2.3 Change of Operad
15(1)
1.3 Trees
16(4)
1.4 Alternative Definitions for Operads
20(3)
1.5 Free Operads
23(3)
1.6 The Tensor Product of Operads
26(3)
1.7 The Boardman-Vogt Resolution of an Operad
29(8)
1.8 Configuration Spaces and the Fulton--MacPherson Operad
37(7)
1.9 Configuration Spaces and the Operad of Little Cubes
44(5)
2 Simplicial Sets
49(42)
2.1 The Simplex Category Δ
49(4)
2.2 Simplicial Sets and Geometric Realization
53(4)
2.3 The Geometric Realization as a Cell Complex
57(5)
2.4 Simplicial Sets as a Category of Presheaves
62(8)
2.5 Products of Simplicial Sets and Shuffle Maps
70(7)
2.6 Simplicial Spaces and Bisimplicial Sets
77(3)
2.6.1 Simplicial Spaces
77(1)
2.6.2 Bisimplicial Sets
78(2)
2.7 Simplicial Categories and Simplicial Operads
80(11)
2.7.1 Internal Versus Enriched Categories and Operads
81(1)
2.7.2 Simplicial Categories
82(1)
2.7.3 Boardman--Vogt Resolution
83(2)
2.7.4 Homotopy--Coherent Nerve
85(1)
2.7.5 Simplicial Operads
86(1)
2.7.6 The Barratt--Eccles Operad
87(2)
2.7.7 The Simplicial Boardman--Vogt Resolution of an Operad
89(2)
3 Dendroidal Sets
91(42)
3.1 Trees
91(4)
3.2 The Category Ω of Trees
95(3)
3.3 Faces and Degeneracies in Ω
98(8)
3.3.1 Outer Faces
99(1)
3.3.2 Inner Faces
99(1)
3.3.3 Degeneracies
100(1)
3.3.4 Codendroidal Identities
100(1)
3.3.5 Factorization of Morphisms Between Trees
101(3)
3.3.6 Some Limits and Colimits in Ω
104(2)
3.4 Dendroidal Sets
106(6)
3.5 Categories Related to Dendroidal Sets
112(7)
3.5.1 Dendroidal Sets and Operads
113(1)
3.5.2 Dendroidal Sets and Simplicial Sets
114(1)
3.5.3 Dendroidal Sets and Simplicial Operads
115(1)
3.5.4 Open Dendroidal Sets
115(1)
3.5.5 Closed Dendroidal Sets
116(1)
3.5.6 Uncoloured Dendroidal Sets
117(1)
3.5.7 Dendroidal Sets and Γ-Sets
118(1)
3.6 Normal Dendroidal Sets and Skeletal Filtration
119(5)
3.7 Normal Monomorphisms and Normalization
124(9)
4 Tensor Products of Dendroidal Sets
133(28)
4.1 Elementary Properties and Shuffles of Trees
133(11)
4.2 The Tensor Product of a Simplicial and a Dendroidal Set
144(2)
4.3 Tensor Products and Normal Monomorphisms
146(9)
4.4 Unbiased Tensor Products
155(6)
5 Kan Conditions for Simplicial Sets
161(50)
5.1 Kan Complexes and ∞-Categories
161(7)
5.2 Fibrations Between Simplicial Sets
168(5)
5.3 Saturated Classes and Anodyne Morphisms
173(6)
5.4 Products, Joins, and Spines of Simplices
179(7)
5.5 Fibrations Between Mapping Spaces
186(5)
5.6 Equivalences in ∞-Categories
191(8)
5.7 Minimal ∞-Categories and Minimal Kan Complexes
199(7)
5.8 Minimal Fibrations Between ∞-Categories
206(5)
6 Kan Conditions for Dendroidal Sets
211(54)
6.1 Dendroidal Kan Complexes and ∞-Operads
211(9)
6.2 Fibrations and Anodyne Morphisms Between Dendroidal Sets
220(7)
6.3 Tensor Products and Anodyne Morphisms
227(13)
6.4 Fibrations Between Mapping Spaces of Dendroidal Sets
240(2)
6.5 Spines and Leaves of Trees
242(5)
6.6 Joins of Trees
247(5)
6.7 Equivalences in ∞-Operads
252(3)
6.8 Minimal Fibrations Between ∞-Operads
255(10)
Part II The Homotopy Theory of Simplicial and Dendroidal Sets
7 Model Categories
265(38)
7.1 Axioms for a Model Category
266(4)
7.2 Some Background on Topological Spaces
270(5)
7.3 A Model Structure for Topological Spaces
275(5)
7.4 Homotopies Between Morphisms in a Model Category
280(7)
7.5 The Homotopy Category of a Model Category
287(3)
7.6 Brown's Lemma and Proper Model Categories
290(3)
7.7 Transfer of Model Structures
293(4)
7.8 Homotopy Pushouts and the Cube Lemma
297(6)
8 Model Structures on the Category of Simplicial Sets
303(50)
8.1 The Categorical Model Structure on Simplicial Sets
304(9)
8.2 The Kan--Quillen Model Structure on Simplicial Sets
313(5)
8.3 Quillen Adjunctions and Derived Functors
318(12)
8.4 Homotopy Groups of Simplicial Sets
330(7)
8.5 Geometric Realizations and Fibrations
337(4)
8.6 The Equivalence Between Simplicial Sets and Topological Spaces
341(2)
8.7 Categorical Weak Equivalences Between ∞-Categories
343(4)
8.8 The Covariant Model Structure
347(6)
9 Three Model Structures on the Category of Dendroidal Sets
353(70)
9.1 The A-Model Structure for Dendroidal Sets
354(17)
9.2 The Operadic Model Structure
371(7)
9.3 Open and Uncoloured Dendroidal Sets
378(5)
9.4 The Relative A-Model Structure
383(4)
9.5 The Covariant Model Structure on Dendroidal Sets
387(9)
9.6 The Absolute Covariant Model Structure
396(8)
9.7 The Picard Model Structure
404(19)
Part III The Homotopy Theory of Simplicial and Dendroidal Spaces
10 Reedy Categories and Diagrams of Spaces
423(30)
10.1 Reedy Categories
423(3)
10.2 Reedy Fibrations
426(5)
10.3 The Reedy Model Structure
431(4)
10.4 Simplicial Objects and Geometric Realization
435(7)
10.5 Homotopy Colimits
442(5)
10.6 A Version of Quillen's Theorem B
447(6)
11 Mapping Spaces and Bousfield Localizations
453(28)
11.1 Mapping Spaces
453(7)
11.2 Common Models for Mapping Spaces
460(7)
11.3 Left Bousfield Localizations
467(2)
11.4 Existence of Left Bousfield Localizations
469(5)
11.5 Localizable Sets of Morphisms
474(7)
12 Dendroidal Spaces and ∞-Operads
481(42)
12.1 Dendroidal Segal Spaces
482(9)
12.2 Complete Dendroidal Segal Spaces
491(7)
12.3 Complete Weak Equivalences
498(7)
12.4 The Tensor Product of Dendroidal Spaces
505(4)
12.5 Closed Dendroidal Spaces
509(5)
12.6 Reduced Dendroidal Spaces
514(5)
12.7 Simplicial Spaces
519(4)
13 Left Fibrations and the Covariant Model Structure
523(32)
13.1 The Covariant Model Structure on Dendroidal Spaces
524(6)
13.2 Simplicial Systems of Model Categories
530(10)
13.3 Homotopy Invariance of the Covariant Model Structure
540(4)
13.4 The Homotopy Theory of Algebras
544(5)
13.5 Algebras and Left Fibrations
549(6)
14 Simplicial Operads and ∞-Operads
555(36)
14.1 Simplicial Categories with Fixed Objects
556(3)
14.2 Equivalences in Simplicial Categories
559(5)
14.3 A Model Structure for Simplicial Operads
564(3)
14.4 The Sparse Model Structure
567(6)
14.5 Simplicial Operads and Dendroidal Spaces
573(3)
14.6 The Homotopy-Coherent Nerve
576(5)
14.7 Operads with a Single Colour
581(6)
14.8 Algebras for ∞-Operads and for Simplicial Operads
587(4)
Epilogue 591(10)
References 601(6)
Index 607
Both authors are experienced researchers in the field who have contributed significantly to the development of the theory contained in this book. They have lectured extensively on simplicial and dendroidal sets.