标题： 可微结构在两个开集的并集上 显示英文标题

标题： Differentiable structures on a union of two open sets

摘要： 在最近的一篇论文中，作者对非豪斯多夫一维流形$\mathbb{L}$的微分结构进行了分类，该流形称为带有两个原点的直线，它是通过将两个实数线的副本$\mathbb{R}$通过$\mathbb{R}\setminus 0$的恒等同胚映射粘合而成的。 在这里，我们给出了另一种非豪斯多夫一维流形$\mathbb{Y}$（称为字母“$Y$”）的微分结构分类，该流形是通过将两个$\mathbb{R}$的副本通过正实数的恒等映射粘合而成的。 结果表明，与实数轴不同，对于每个$r=1,\ldots,\infty$，流形$\mathbb{L}$和$\mathbb{Y}$都允许不可数多个两两不同胚的$\mathcal{C}^{k}$结构。 我们还观察到这些分类的证明非常相似。 这使得可以形式化这些论点并将其扩展到关于任意范畴中箭头的某种一般性陈述。 显示英文摘要

摘要： In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold $\mathbb{L}$ called the line with two origins which is obtained by gluing two copies of the real line $\mathbb{R}$ via the identity homeomorphism of $\mathbb{R}\setminus 0$. Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold $\mathbb{Y}$ (called letter "$Y$") obtained by gluing two copies of $\mathbb{R}$ via the identity map of positive reals. It turns out that, in contrast to the real line, for every $r=1,\ldots,\infty$, both manifolds $\mathbb{L}$ and $\mathbb{Y}$ admit uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures. We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories.