论文标题
部分可观测时空混沌系统的无模型预测
Diffeological submanifolds and their friends
论文作者
论文摘要
平滑的歧管托有不同类型的子延伸物，包括嵌入式，弱插入和沉浸的亚曼叶。沉浸式亚曼叶的概念需要额外的结构（即选择拓扑）；当这种额外的结构唯一时，我们将子集称为独特的沉浸式子手机。差异学提供了submanifold的另一个内在概念：一个差异下降。 我们表明，从分类的角度来看，差异从其他角度上升出来：视图作为集合类别的具体类别，最初的形态恰好是（差异）诱导，这是具有差异学亚法的差异性。此外，如果我们将歧管视为拓扑空间类别的具体类别，我们将恢复Joris和Preissmann的伪象征概念。 我们表明这些概念都是不同的。特别是，乔里斯（Joris）的定理产生了一个差异学的submanifold，其包含不是沉浸式的，回答了伊格莱斯西亚斯 - 泽莫尔（Iglesias-Zemmour）提出的一个问题。我们还将局部感应描述为那些局部注入性的伪放入。 在附录中，我们回顾了乔里斯定理的证明，指出了文献中发生的其他几个证据之一的缺陷，并说明了Submanifolds如何从其周围的歧管中继承paracrocactness。
A smooth manifold hosts different types of submanifolds, including embedded, weakly-embedded, and immersed submanifolds. The notion of an immersed submanifold requires additional structure (namely, the choice of a topology); when this additional structure is unique, we call the subset a uniquely immersed submanifold. Diffeology provides yet another intrinsic notion of submanifold: a diffeological submanifold. We show that from a categorical perspective diffeology rises above the others: viewing manifolds as a concrete category over the category of sets, the initial morphisms are exactly the (diffeological) inductions, which are the diffeomorphisms with diffeological submanifolds. Moreover, if we view manifolds as a concrete category over the category of topological spaces, we recover Joris and Preissmann's notion of pseudo-immersions. We show that these notions are all different. In particular, a theorem of Joris from 1982 yields a diffeological submanifold whose inclusion is not an immersion, answering a question that was posed by Iglesias-Zemmour. We also characterize local inductions as those pseudo-immersions that are locally injective. In appendices, we review a proof of Joris' theorem, pointing at a flaw in one of the several other proofs that occur in the literature, and we illustrate how submanifolds inherit paracompactness from their ambient manifold.
