论文标题
公制空间和同喻类型
Metric spaces and homotopy types
论文作者
论文摘要
通过类似于Spivak的方法，有一个实现函数，它将持久性图$ y $用于简单集的$ y $用于扩展的伪金属空间（或ep-metric Space）$ re（y）$。函子$ re $具有正确的伴随，称为单数函子，将ep-metric Space $ z $带到持久图$ s（z）$。我们给出了$ re（y）$的明确说明，并证明它仅取决于$ 1 $ - skeleton $ sk_ {1} y $ $ y $。如果$ x $是一个完全有序的ep米型空间，那么在实现vietoris-rips图$ v _ {\ ast}（x）（x）（x）$和ep-metric space $ x $之间的实现之间，有一个同构$ re（v _ {\ ast}（x）（x）（x）（x）（x）（x）（x））\ x $。持久图$ v _ {\ ast}（x）$和$ s（x）$对于所有此类$ x $都是等效的。
By analogy with methods of Spivak, there is a realization functor which takes a persistence diagram $Y$ in simplicial sets to an extended pseudo-metric space (or ep-metric space) $Re(Y)$. The functor $Re$ has a right adjoint, called the singular functor, which takes an ep-metric space $Z$ to a persistence diagram $S(Z)$. We give an explicit description of $Re(Y)$, and show that it depends only on the $1$-skeleton $sk_{1}Y$ of $Y$. If $X$ is a totally ordered ep-metric space, then there is an isomorphism $Re(V_{\ast}(X)) \cong X$, between the realization of the Vietoris-Rips diagram $V_{\ast}(X)$ and the ep-metric space $X$. The persistence diagrams $V_{\ast}(X)$ and $S(X)$ are sectionwise equivalent for all such $X$.
