Handbook of Homotopy Theory

Haynes Miller

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Print ISBN: 9780815369707
• eBook ISBN: 9781351251600
• Edition: 1
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories.

The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

1

Goodwillie calculus

Gregory Arone and Michael Ching

Mathematics Subject Classification. 55P65, 55P48, 55P42, 19D10.

Key words and phrases. calculus of functors, identity functor, operads, algebraic K-theory.

Goodwillie calculus is a method for analyzing functors that arise in topology. One may think of this theory as a categorification of the classical differential calculus of Newton and Leibnitz, and it was introduced by Tom Goodwillie in a series of foundational papers [44, 45, 46].

The starting point for the theory is the concept of an n-excisive functor, which is a categorification of the notion of a polynomial function of degree n. One of Goodwillie’s key results says that every homotopy functor F has a universal approximation by an n-excisive functor P_nF, which plays the role of the n-th Taylor approximation of F. Together, the functors P_nF fit into a tower of approximations of F: the Taylor tower

F⟶⋯⟶PnF⟶⋯⟶P1F⟶P0F.

It turns out that 1-excisive functors are the ones that represent generalized homology theories (roughly speaking). For example, if F = I is the identity functor on the category of based spaces, then P₁I is the functor P₁I(X) ≃ Ω^∞Σ^∞X. This functor represents stable homotopy theory in the sense that π∗(P1I(X))≅π∗s(X). Informally, this means that the best approximation to the homotopy groups by a generalized homology theory is given by the stable homotopy groups. The Taylor tower of the identity functor then provides a sequence of theories, satisfying higher versions of the excision axiom, that interpolate between stable and unstable homotopy.

The analogy between Goodwillie calculus and ordinary calculus reaches a surprising depth. To illustrate this, let D_nF be the homotopy fiber of the map P_nF → P_n−1F. The functors D_nF are the homogeneous pieces of the Taylor tower. They are controlled by Taylor “coefficients” or derivatives of F. This means that for each n there is a spectrum with an action of Σ_n that we denote ∂_nF, and there is an equivalence of functors

DnF(X)≃Ω∞(∂nF∧X∧n)hΣn.

Here for concreteness F is a homotopy functor from the category of pointed spaces to itself; similar formulas apply for functors to and from other categories. The spectrum ∂_n F pla...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Contents
• Preface
• 1. Goodwillie calculus
• 2. A factorization homology primer
• 3. Polyhedral products and features of their homotopy theory
• 4. A guide to tensor-triangular classification
• 5. Chromatic structures in stable homotopy theory
• 6. Topological modular and automorphic forms
• 7. A survey of models for (∞,n)-categories
• 8. Persistent homology and applied homotopy theory
• 9. Algebraic models in the homotopy theory of classifying spaces
• 10. Floer homotopy theory, revisited
• 11. Little discs operads, graph complexes and Grothendieck–Teichmüller groups
• 12. Moduli spaces of manifolds: a user’s guide
• 13. An introduction to higher categorical algebra
• 14. A short course on ∞-categories
• 15. Topological cyclic homology
• 16. Lie algebra models for unstable homotopy theory
• 17. Equivariant stable homotopy theory
• 18. Motivic stable homotopy groups
• 19. En-spectra and Dyer-Lashof operations
• 20. Assembly maps
• 21. Lubin-Tate theory, character theory, and power operations
• 22. Unstable motivic homotopy theory
• Index