On tested Bousfield-Friedlander localizations

TL;DR

This paper shows that Bousfield-Friedlander localizations with test morphisms can be viewed as left Bousfield localizations, leading to a unified framework that connects various calculi in homotopy theory.

Contribution

It establishes a new characterization of Bousfield-Friedlander localizations as left Bousfield localizations, unifying different calculus frameworks in homotopy theory.

Findings

Homogeneous functors in discrete calculus match those in Goodwillie calculus up to homotopy.

Polynomial model structure of Weiss calculus is an instance of tested Bousfield-Friedlander localization.

A homogeneous model structure can be associated with any calculus from Bousfield-Friedlander localization.

Abstract

We demonstrate that a Bousfield-Friedlander localization with a set of test morphisms in the sense introduced by Bandklayder, Bergner, Griffiths, Johnson, and Santhanam can also be characterized as a left Bousfield localization at the set of test morphisms. This viewpoint enables us to establish a homogeneous model structure associated with any calculus arising from a Bousfield-Friedlander localization of this form. As a corollary, we show that homogeneous functors in discrete calculus coincide up to homotopy with those in Goodwillie calculus. Finally, we illustrate this framework by proving that the polynomial model structure of Weiss calculus is a particular instance of tested Bousfield-Friedlander localization.