Quasi-coherent sheaves in differential geometry

TL;DR

This paper establishes a monoidal model structure on the category of simplicial complete bornological spaces over the real numbers, enabling a homotopical approach to modules over simplicial $C^$-rings in differential geometry.

Contribution

It introduces a new monoidal model structure on simplicial complete bornological spaces and applies it to simplicial $C^$-rings, linking homotopical algebra with differential geometry.

Findings

Category of simplicial complete bornological spaces has a monoidal model structure.

Weak equivalences between monoids induce Quillen equivalences between module categories.

Pre-compact bornology functor preserves and reflects weak equivalences in simplicial $C^$-rings.

Abstract

It is proved that the category of simplicial complete bornological spaces over $\mathbb R$ carries a combinatorial monoidal model structure satisfying the monoid axiom. For any commutative monoid in this category the category of modules is also a monoidal model category with all cofibrant objects being flat. In particular, weak equivalences between these monoids induce Quillen equivalences between the corresponding categories of modules. On the other hand, it is also proved that the functor of pre-compact bornology applied to simplicial $C^\infty$-rings preserves and reflects weak equivalences, thus assigning stable model categories of modules to simplicial $C^\infty$-rings.