We develop the theory of smooth principal bundles for a smooth group G, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define D-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a D-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor’s spaces EG and BG, show that EG → BG is a D-numerable principal bundle, and prove that it classifies all D-numerable principal bundles over any diffeological space. We deduce analogous classification results for D-numerable diffeological bundles and vector bundles.

我们使用差分空间的框架，发展了一个光滑群G的光滑主束的理论。在给出表明为什么不能对任意主束分类的新示例之后，我们定义了D可数束，这是从拓扑结构中可数束的平滑类似物，并证明沿着平滑的同位图拉回D可数束会产生同构回调。然后，我们在Milnor空间EG和BG上定义光滑结构，证明EG → BG是D个可数主束，并证明它对所有D分类在任何差分空间上的无数主体束。我们推导了D个数差论束和向量束的相似分类结果。