Goodwillie approximations to higher categories

TL;DR

This paper develops a framework of Goodwillie towers for infinity-categories, providing new models for homotopy theories of spaces and spectra, with applications to rational and p-local homotopy theory.

Contribution

It constructs a general class of Goodwillie towers for infinity-categories and classifies them via derivatives of the identity functor, connecting to homotopy theories of spaces and spectra.

Findings

Provides a model for simply-connected spaces using coalgebras with Tate diagonals.

Simplifies classification in cases with vanishing Tate cohomology.

Relates homotopy theory of p-local spaces to algebras over spectral Lie operad.

Abstract

We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of infinity-categories C. We classify such Goodwillie towers in terms of the derivatives of the identity functor of C. As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p-local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching's spectral version of the Lie operad. This is a close analogue of Quillen's results on rational homotopy. In the sequel to this paper we work out consequences for the study of $v_n$-periodic unstable homotopy theory and the Bousfield-Kuhn functors.