衍生的光滑流形
Derived Smooth Manifolds
摘要：
我们定义了一个单纯范畴，叫做导流形范畴。它包含了光滑流形范畴作为一个完全离散的子范畴，并且在流形上取任意交时是闭的。一个导流形是一个带有局部$C^\infty$ -环层的空间，该层是通过将欧几里得空间上的光滑函数的同伦零集拼接在一起而得到的。我们证明了导出流形具有稳定的法丛，并且可以嵌入到欧氏空间中。我们定义了一个上同调理论，称为导出配边理论，并使用Pontrjagin-Thom参数，以表明导出配边理论是同构的经典配边理论。这使我们能够定义基本类的配边所有衍生流形。特别地，在我们的理论中，子流形$A,B\subset X$的交$A\cap B$存在于范畴水平上，并且杯积公式$$[A]\smile[B]=[A\cap B]$$成立，即使子流形不是横截的。因此，我们可以把导出流形的理论看作是相交理论的{ \em分类}。
英文摘要：
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.
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