论文标题
无限维$ c^{\ infty} $的平滑同质性 - 歧管
Smooth Homotopy of Infinite-Dimensional $C^{\infty}$-Manifolds
论文作者
论文摘要
在本文中，我们使用同位代数（或抽象同位方法）来研究无限维$ c^\ infty $ - manifolds在方便的演算中的平滑同位素问题。更确切地说，我们讨论地图，部分，主束和规范变换的平滑。 我们首先介绍了遗传性$ c^\ infty $ - paraccroctness，以及在$ c^\ infty $ -Manifold上的半经典条件，这使我们能够在本地参数中使用本地凸度。 Then, we prove that for $C^\infty$-manifolds $M$ and $N$, the smooth singular complex of $C^\infty(M,N)$ is weakly equivalent to the ordinary singular complex of $\mathcal{C}^0(M,N)$ under the hereditary $C^\infty$-paracompactness and semiclassicality conditions on $M$.接下来，我们将此结果推广到$ C^\ infty $ -manifold $ m $在相同条件下的$ m $ $ m $的部分。此外，我们在$ m $上的平稳$ g $捆绑的简单组与$ g $ g $ g $ g $ g $ m $ a $ g $的$ g $ - $ g $与$ c^\ infty $ mmanifold $ -Manifold $ m $之间的$ M $划分的$ m $上的简单类别之间建立了dwyer-kan等价性。 对于证据，我们完全忠实地将$ c^\ infty $ - manifolds的$ c^\ infty $添加到差异空间的类别$ \ mathcal {d} $中，并通过模型类别$ $ $ \ nathcal cagory and and and and and and and d} $ \ d} d} d} d}的平滑差异空间理论。 $ \ Mathcal {C}^0 $的弧生成空间。然后，$ m $的遗传性$ c^\ infty $ - paraccrompactness和半经典性条件意味着$ m $具有$ \ mathcal {d} $中的同型同源物类型。该结果可以被视为对Milnor，Palais和Heisey在无限二维拓扑歧管的同型类型上的结果进行平滑的改进。
In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional $C^\infty$-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations. We first introduce the notion of hereditary $C^\infty$-paracompactness along with the semiclassicality condition on a $C^\infty$-manifold, which enables us to use local convexity in local arguments. Then, we prove that for $C^\infty$-manifolds $M$ and $N$, the smooth singular complex of $C^\infty(M,N)$ is weakly equivalent to the ordinary singular complex of $\mathcal{C}^0(M,N)$ under the hereditary $C^\infty$-paracompactness and semiclassicality conditions on $M$. We next generalize this result to sections of fiber bundles over a $C^\infty$-manifold $M$ under the same conditions on $M$. Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal $G$-bundles over $M$ and that of continuous principal $G$-bundles over $M$ for a Lie group $G$ and a $C^\infty$-manifold $M$ under the same conditions on $M$, encoding the smoothing results for principal bundles and gauge transformations. For the proofs, we fully faithfully embed the category $C^\infty$ of $C^\infty$-manifolds into the category $\mathcal{D}$ of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category $\mathcal{D}$ and the model category $\mathcal{C}^0$ of arc-generated spaces. Then, the hereditary $C^\infty$-paracompactness and semiclassicality conditions on $M$ imply that $M$ has the smooth homotopy type of a cofibrant object in $\mathcal{D}$. This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey on the homotopy type of infinite-dimensional topological manifolds.
