Derived Smooth Manifolds
Abstract
We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local $C^\infty$rings that is obtained by patching together homotopy zerosets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a PontrjaginThom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection $A\cap B$ of submanifolds $A,B\subset X$ exists on the categorical level in our theory, and a cup product formula $$[A]\smile[B]=[A\cap B]$$ holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a {\em categorification} of intersection theory.
 Publication:

Ph.D. Thesis
 Pub Date:
 October 2008
 arXiv:
 arXiv:0810.5174
 Bibcode:
 2008PhDT.......449S
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 55N22 (Primary);
 55N33;
 18F20 (Secondary);
 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 55N22 (Primary);
 55N33;
 18F20 (Secondary)
 EPrint:
 57 pages. Reformulation of author's PhD thesis. To appear in Duke Math J.