The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces

TL;DR

This paper develops an $R$-local homotopy theory for smooth spaces modeled by simplicial presheaves, establishing Quillen equivalences and fibrant replacements that connect smooth spaces with their singular complexes.

Contribution

It introduces an $R$-local model structure for smooth spaces, proves its equivalence with a motivic-style localization, and relates the singular complex functor to fibrant replacements and concordance sheaves.

Findings

The $R$-localization of smooth spaces matches a motivic-style localization.

The singular complex functor is a Quillen equivalence up to weak equivalences.

A functorial fibrant replacement for $R$-local smooth spaces is constructed.

Abstract

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. There is a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We consider the localisation of the (projective or injective) model category of smooth spaces at the morphisms which become weak equivalences under the singular complex functor. We prove that this localisation agrees with a motivic-style $\mathbb{R}$-localisation of the model category of smooth spaces. Further, we exhibit the singular complex functor for smooth spaces as one of several Quillen equivalences between model categories for spaces and the above $\mathbb{R}$-local model category of smooth spaces. In the process, we show that the singular complex functor agrees with the homotopy colimit functor up to a natural zig-zag of weak equivalences. We provide a functorial fibrant replacement in the $\mathbb{R}$-local model category of smooth spaces and use this to compute mapping spaces in terms of singular complexes. Finally, we explain the relation of our fibrant replacement to the concordance sheaf construction introduced recently by Berwick-Evans, Boavida de Brito and Pavlov.