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

Severin Bunk

Published 2020-07-12Version 1

Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. We define a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We exhibit this functor as one of several Quillen equivalences between the Kan-Quillen model category of simplicial sets and a motivic-style $\mathbb{R}$-localisation of the (projective or injective) model category of smooth spaces. These Quillen equivalences and their interrelations are powerful tools: for instance, they allow us to give a purely homotopy-theoretic proof of a Whitehead Approximation Theorem for manifolds. We show that an adapted version of the concordance sheaf construction of Berwick-Evans, de Brito, and Pavlov provides a fibrant replacement functor in the $\mathbb{R}$-local model category of smooth spaces. This leads to an independent and simplified proof of one of their main results: the smooth singular complex functor preserves the homotopy type of mapping spaces. Finally, we show that the $\mathbb{R}$-local model category of smooth spaces formalises the homotopy theory on sheaves used by Galatius, Madsen, Tillmann, and Weiss in their seminal paper on the homotopy type of the cobordism category.