论文标题
$ \ mathbb {r} $ - 平滑空间的本地同义理论
The $\mathbb{R}$-Local Homotopy Theory of Smooth Spaces
论文作者
论文摘要
笛卡尔空间上的简单预选者提供了光滑空间的一般概念。有一个相应的平滑版本的单数复合函数，该版本将光滑的空间映射到简单的集合。我们考虑在形态上平滑空间的（投影或注入性）模型类别的定位，这些模型在形态上变成了奇异复杂函数下的弱等价。我们证明，此本地化与动机风格的$ \ mathbb {r} $ - 平滑空间模型类别的本地化。此外，我们展示了平滑空间的单数复合函数，这是空间模型类别与上述$ \ m athbb {r} $之间的几个quillen等效函数之一 - 平滑空间的本地模型类别。在此过程中，我们表明，奇异的复合函子与同质函数的同型函数一致，直到弱等效性的自然曲折。我们在$ \ mathbb {r} $ - 光滑空间的本地模型类别中提供功能纤维替代品，并使用此功能以单数复合物来计算映射空间。最后，我们解释了我们的原始替代品与Berwick-Evans，Boavida de Brito和Pavlov最近提出的一致性捆捆结构的关系。
Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. There is a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We consider the localisation of the (projective or injective) model category of smooth spaces at the morphisms which become weak equivalences under the singular complex functor. We prove that this localisation agrees with a motivic-style $\mathbb{R}$-localisation of the model category of smooth spaces. Further, we exhibit the singular complex functor for smooth spaces as one of several Quillen equivalences between model categories for spaces and the above $\mathbb{R}$-local model category of smooth spaces. In the process, we show that the singular complex functor agrees with the homotopy colimit functor up to a natural zig-zag of weak equivalences. We provide a functorial fibrant replacement in the $\mathbb{R}$-local model category of smooth spaces and use this to compute mapping spaces in terms of singular complexes. Finally, we explain the relation of our fibrant replacement to the concordance sheaf construction introduced recently by Berwick-Evans, Boavida de Brito and Pavlov.
