Every principal $G$-bundle over $X$ is classified up to equivalence by a homotopy class $X o BG$, where $BG$ is the classifying space of $G$. On the other hand, for every nice topological space $X$ Milnor constructed a strict model of its loop space $ilde{Omega}X$, that is a group. Moreover, the morphisms of topological groups $ilde{Omega}X o G$ generate all the $G$-bundles over $X$ up to equivalence. In this paper, we show that the relation between Milnor’s loop space and the classifying space functor is, in a precise sense, an adjoint pair between based spaces and topological groups in a homotopical context. This proof leads to a classification of principal bundles over a fixed space, that is dual to the classification of bundles with a fixed group. Such a result clarifies the deep relation that exists between the theory of bundles, the classifying space construction and the loop space, which are very important in topological $K$-theory, group cohomology, and homotopy theory.
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机译:每一个超过$ X $的本金$ G $捆绑都由同构类$ X 归类为BG $,其中$ BG $是$ G $的分类空间。另一方面,对于每个不错的拓扑空间$ X $,Milnor都会为其循环空间$ tilde { Omega} X $构造一个严格的模型,即一个组。此外,拓扑组$ tilde { Omega} X to G $的态射会生成超过$ X $的所有$ G $束,直到等价为止。在本文中,我们证明了Milnor的循环空间和分类空间函子之间的关系在精确意义上是同位上下文中基础空间和拓扑组之间的伴随对。该证明导致在固定空间上对主捆绑包进行分类,这与对具有固定组的捆绑包进行分类是双重的。这样的结果阐明了束理论,分类空间构造和环空间之间存在的深层关系,这在拓扑$ K $理论,群同调学和同伦理论中非常重要。
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