Derived Smooth Manifolds

TL;DR

This paper introduces derived manifolds as a new category extending smooth manifolds, enabling intersection theory to be categorified and defining a cobordism theory that aligns with classical results.

Contribution

It defines derived manifolds with a sheaf of smooth functions, proves their embeddability into Euclidean space, and establishes a cobordism theory equivalent to classical cobordism.

Findings

Derived cobordism is isomorphic to classical cobordism.

Intersection of submanifolds exists categorically without transversality.

Derived manifolds enable a categorified intersection theory.

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 zero-sets 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 Pontrjagin-Thom 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.