Goodwillie Approximations To Higher Categories
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Goodwillie Approximations to Higher Categories
Author: Gijs Heuts
language: en
Publisher: American Mathematical Society
Release Date: 2021-11-16
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Goodwillie Approximations to Higher Categories
Goodwillie calculus involves the approximation of functors between higher categories by so-called polynomial functors. We show (under mild hypotheses) how to associate to a higher category C a Goodwillie tower, consisting of categories which are polynomial in an appropriate sense. These polynomial approximations enjoy universal properties with respect to polynomial functors out of C. Furthermore, we provide a classification of such Goodwillie towers in terms of the stabilization of C and the derivatives of the identity functor. In special cases this classification becomes very simple, allowing us to draw conclusions about the structure of the category C. As an example we give an application to Quillen's rational homotopy theory. In the sequel to this paper we work out consequences for the study of vn-periodic unstable homotopy theory and the Bousfield-Kuhn functors.
Handbook of Homotopy Theory
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.