This book provides an informal and geodesic introduction to factorization homology, focusing on providing intuition through simple examples. Along the way, the reader is also introduced to modern ideas in homotopy theory and category theory, particularly as it relates to the use of infinity-categories. As with the original lectures, the text is meant to be a leisurely read suitable for advanced graduate students and interested researchers in topology and adjacent fields.
Author(s): Hiro Lee Tanaka
Series: SpringerBriefs in Mathematical Physics, 39
Publisher: Springer
Year: 2021
Language: English
Pages: 84
Preface
Acknowledgements
Contents
1 The One-Dimensional Case
1.1 The Algebra of Disks
1.2 The Co-center: A First Stab at a Circle Invariant
1.3 (infty,1)-Categories, a First Pass
1.4 Algebra of Disks, Revisited
1.5 Factorization Homology of the Circle
References
2 Interlude on (infty,1)-Categories
2.1 Homotopy Equivalences, and Equivalences of (infty,1)-Categories
2.2 Contractibility
2.3 Combinatorial Models and (infty,1)-Categories: Simplicial Sets
2.4 Combinatorial Models and (infty,1)-Categories: Weak Kan Complexes
2.4.1 Nerves
2.4.2 Some Useful Constructions
2.5 Exercises
References
3 Factorization Homology in Higher Dimensions
3.1 Review of Last Talk (Chap. 1摥映數爠eflinkchapter.dim111)
3.2 The Example of Hochschild Chains
3.3 The Algebra of Disks in Higher Dimensions
3.3.1 Framings and Orientations, a First Glance
3.3.2 mathbbEn-Algebras
3.3.3 Examples of mathbbEn-Algebras
3.4 Framings and Tangential G-Structures
3.5 Factorization Homology
3.5.1 Examples
3.6 Leftovers and Elaborations
3.6.1 What are Left Kan Extensions?
3.6.2 What's Up with Sifted Colimits?
3.6.3 How Many Excisive Theories are There?
3.6.4 Locally Constant Factorization Algebras
3.6.5 How Good a Manifold Invariant is Factorization Homology?
3.6.6 Are We Stuck with Algebras Only?
3.7 Exercises
References
4 Topological Field Theories and the Cobordism Hypothesis
4.1 Review of Last Lecture (Chap. 3摥映數爠eflinkchapter.facthominhigherdims33)
4.2 Cobordisms and Higher Categories
4.3 The Cobordism Hypothesis
4.4 The Cobordism Hypothesis in Dimension 1, for Vector Spaces
4.5 Full Dualizability
4.6 The Point is Fully Dualizable
4.7 Factorization Homology as a Topological Field Theory
4.8 Leftovers and Elaborations
4.8.1 The Cobordism Categories
4.8.2 Framings
4.8.3 Structure Groups and Homotopy Fixed Points
4.9 Exercises
References
Appendix Index
Index
