Simplicial and Dendroidal Homotopy Theory

Gijs Heuts, Ieke Moerdijk (Autoren)

Buch | Hardcover

XX, 612 Seiten

2022 | 1st ed. 2022
Springer International Publishing (Verlag)
978-3-031-10446-6 (ISBN)

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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.

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.

Part I The Elementary Theory of Simplicial and Dendroidal Sets.- 1 Operads.- 2 Simplicial Sets.- 3 Dendroidal Sets.- 4 Tensor Products of Dendroidal Sets.- 5 Kan Conditions for Simplicial Sets.- 6 Kan Conditions for Dendroidal Sets.- Part II The Homotopy Theory of Simplicial and Dendroidal Sets.- 7 Model Categories.- 8 Model Structures on the Category of Simplicial Sets.- 9 Three Model Structures on the Category of Dendroidal Sets.- Part III The Homotopy Theory of Simplicial and Dendroidal Spaces.- 10 Reedy Categories and Diagrams of Spaces.- 11 Mapping Spaces and Bousfield Localizations.- 12 Dendroidal Spaces and -Operads.- 13 Left Fibrations and the Covariant Model Structure.- 14 Simplicial Operads and -Operads.- Epilogue.- References.- Index.

"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)

Erscheinungsdatum	04.08.2022
Reihe/Serie	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Zusatzinfo	XX, 612 p. 1 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	1105 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
	Mathematik / Informatik ► Mathematik ► Algebra
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Boardman-Vogt • Bousfield localization • dendroidal set • higher algebra • infinity-category • infinity-operad • Model categories • open access • operads • simplicial operad • simplicial set • Simplicial space
ISBN-10	3-031-10446-3 / 3031104463
ISBN-13	978-3-031-10446-6 / 9783031104466
Zustand	Neuware