Quasi-coherent sheaves in differential geometry Dennis Borisov Kobi Kremnizer University of Goettingen, Germany University of Oxford, UK Bunsenstr. 3-4, 37073 Göttingen Woodstock Rd, Oxford OX2 6GG
[email protected] [email protected] July 31, 2017 Abstract It is proved that the category of simplicial complete bornological spaces over R carries a combinatorial monoidal model structure satisfy- ing 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 correspond- ing categories of modules. On the other hand, it is also proved that the functor of pre-compact bornology applied to simplicial C∞-rings preserves and reflects weak equivalences, thus assigning stable model categories of modules to simplicial C∞-rings. MSC codes: 58A05, 46A08 Keywords: C∞-rings, bornological spaces, pre-compact bornology, quasi-coherent sheaves, quasi-abelian categories Contents 1 Introduction 2 arXiv:1707.01145v2 [math.DG] 28 Jul 2017 2 The category of complete bornological R-spaces 5 2.1 Quasi-abelian structure . 6 2.2 Projective objects and monoidal structure . 9 2.3 Locally separable spaces . 12 2.4 Local presentability and small objects . 15 2.5 Modelstructures ......................... 19 2.6 Cofibrant resolutions and flatness . 24 1 3 Bornological rings of C∞-functions 30 3.1 The pre-compact bornology . 30 3.2 Modelstructure.......................... 32 3.3 Quasi-coherent sheaves . 34 1 Introduction Our goal (beyond this paper) is to adapt some very useful techniques from algebraic geometry (e.g.