The original de Rham cohomology due to Souriau and the singular cohomology in diffeology are not isomorphic to each other in general. This manuscript introduces a singular de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. It is also proved that a morphism called the factor map from the original de Rham complex to the new one is a quasi-isomorphism for a manifold and, more general, a space with singularities. Moreover, Chen's iterated integrals are considered in a diffeological framework. As a consequence, we deduce that the bar complex of the original de Rham complex of a simply-connected diffeological space is quasi-isomorphic to the singular de Rham complex of the diffeological free loop space provided the factor map for the underlying diffeological space is a quasi-isomorphism. The process for proving the assertion yields the Leray--Serre spectral sequence and the Eilenberg--Moore spectral sequence in diffeology.

中文翻译：

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差分空间的简单 cochain 代数

由 Souriau 引起的原始 de Rham 上同调和差异学中的单数上同调一般不是彼此同构的。这份手稿介绍了一个奇异的 de Rham 复合体，该复合体具有一个积分映射到奇异的 cochain 复合体中，该复合体给出了每个差异空间的 de Rham 定理。还证明了从原始 de Rham 复合体到新复合体的称为因子映射的态射是流形的准同构，更一般地说，是具有奇点的空间。此外，Chen 的迭代积分是在一个差分框架中考虑的。作为结果，我们推导出简单连接的差分空间的原始 de Rham 复形的 bar 复形与差分自由环空间的奇异 de Rham 复形拟同构，前提是基础差分空间的因子映射是准同构的。证明断言的过程产生了 Leray--Serre 谱序列和 Eilenberg--Moore 谱序列。