The $$\mathbb {R}$$-local homotopy theory of smooth spaces

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