Goodwillie approximations to higher categories

Abstract

Introduction vii Chapter 1. Main results Chapter 2. Constructing n-excisive approximations Chapter 3. Another construction of polynomial approximations Chapter 4. Coalgebras in stable ∞-operads 4.1. Truncations of ∞-operads 4.2. Coalgebras in a corepresentable ∞-operad 4.3. Coalgebras in an n-truncated stable ∞-operad Chapter 5. The space of Goodwillie towers 5.1. The Tate diagonal 5.2. Constructing n-stages 5.3. A classification of n-stages 5.4. The case of vanishing Tate constructions Chapter 6. Examples 6.1. Divided power coalgebras and Koszul duality 6.2. Rational homotopy theory 6.3. Spaces with homotopy groups in a finite range 6.4. Spaces and Tate coalgebras 6.5. Further remarks on the Goodwillie tower of the homotopy theory of spaces Appendix A. Compactly generated ∞-categories Appendix B. Some facts from Goodwillie calculus Appendix C. Truncations C.1. The homotopy theory of truncated ∞-operads C.2. Truncations of stable ∞-operads C.3. A cobar construction for stable ∞-operads C.4. Coalgebras Bibliography iii

Spectral Algebra Models of Unstable $$v_n$$-Periodic Homotopy Theory

“…But what does it mean explicitly to endow a spectrum with a Σ ∞ Ω ∞ -coalgebra structure? This seems to be a difficult question, but Arone, Klein, Heuts, and others have partial information (see [Kle05], [Heu16]). Rationally, however, Σ ∞ Ω ∞ is equivalent (on connected spaces) to the free commutative coalgebra functor, and coalgebras for this comonad are therefore rationally equivalent to commutative coalgebras.…”

Combinatorial parametrised spectra

“…factoring the codomain projection functor ev [1] : C ∆ 1 → C. In this tower, the n-th jet ∞-bundle p n : J n C → C is the result of unstraightening the functor c → P n (C / /c ) that sends c ∈ C to the n-th excisive approximation of the ∞-category C / /c (the general theory of excisive approximations of ∞-categories was worked out by Heuts in [Heu15]). The first excisive approximation corresponds to passing to the stabilisation, and the various excisive approximations fit together into a tower of left adjoints…”

“…Provided that this can be made precise, we expect T ∞ c to play a central role in the deformation theory of c inside C, analogous to the Lie algebras controlling derived deformations problems in derived algebraic geometry over a field of characteristic zero (we recommend Lurie's ICM address [Lur10] for a compelling overview of this perspective on formal moduli problems). This analogy is borne out for the ∞-category of 1-connected rational spaces, where for a 1-connected space X, T ∞ X is the rational Whitehead-Lie algebra [Qui69,Heu15]. We intend to undertake a detailed study of these ideas in future work.…”

Goodwillie's calculus of functors and higher topos theory

“…In [BR14] a reasonably general framework for Goodwillie calculus in the language of model categories is developed. In [Heu15], the author constructs Goodwillie approximations of arbitrary categories. Here, however, we are particularly interested in functor categories, and more specifically, those valued in spaces.…”