On tested Bousfield–Friedlander localizations

Abstract

Abstract We show that a Bousfield–Friedlander localization of a model category of functors with respect to a set of test morphisms, as introduced by Bandklayder, Bergner, Griffiths, Johnson and Santhanam, can be characterized as a left Bousfield localization at a specific tensoring of the set of homotopy generators. This reformulation immediately strengthens their main result by establishing that such localizations always yield cellular or combinatorial model categories. We apply this framework to functor calculus in two key ways. First, we construct a homogeneous model structure for any calculus arising from tested Bousfield–Friedlander localization. As a consequence, we construct a Quillen adjunction between homogeneous functors in Goodwillie calculus and homogeneous functors in the discrete calculus of Bauer, Johnson and McCarthy. Second, we show that the polynomial model structure of Weiss calculus is a specific instance of this localization framework, suggesting a unifying perspective on various flavours of functor calculus.