Derived Smooth Manifolds, Part I

A well-known shortcoming of the category of smooth manifolds is its lack of arbitrary pullbacks. A pullback of manifolds, and in particular an intersection of submanifolds, exists only along maps which are transversal. This problem can be overcome by passing to the larger category of derived smooth manifolds. The construction of this category combines ideas from algebraic geometry, homotopy theory and of course differential topology. We can describe this construction in several steps. Firstly, we consider the relation between manifolds and schemes. Here, we employ the so-called C^\infty-rings, which are algebraic objects encoding the structure of the collection of smooth functions on R^n beyond that of an R-algebra. By the general philosophy of algebraic geometry, their duals give rise to geometric objects, called C^\infty-schemes. These geometric objects are primarily studied as models for synthetic differential geometry. Secondly, we introduce homotopy theory into the picture. This step adapts the ideas of derived algebraic geometry to the setting of C^\infty-schemes. Our approach replaces the algebraic objects involved by their simplicial counterparts. In this context, the main objective is to develop a homotopy theory of presheaves which allows us to work with sheaf axioms 'up to homotopy'. Succinctly, a derived smooth manifold can be described as a homotopical C^\infty scheme of finite type. In my talk, I will highlight some steps of the rather intricate construction described above. Hopefully, this will give the audience a perspective from which to think further about these exciting interactions between algebraic geometry, homotopy theory and differential topology.