论文标题
对角线和交织距离的增厚
Thickening of the diagonal and interleaving distance
论文作者
论文摘要
给定拓扑空间$ x $，增厚的内核是$（\ mathbb {r} _ {\ geq0}，+）$的单型预毛，在$ x $上的派生核的单个类别中，值。在$（\ mathbb {r}，+）$上定义了双重的内核。对于如此增厚的内核，一个人自然会在$ x $上派出的带束带类别的距离。 我们证明，在包含$ 0 $的间隔上定义了一个浓厚的内核，并且是独一无二的，从而使我们可以在两种不同的情况下构造（Bi-）增厚。 首先，当$ x $是一个``好''公制空间时，从对角的小通常增厚开始。相关的交织距离满足稳定性的特性，Lipschitz内核产生了Lipschitz地图。 其次，通过使用[GKS12]，当$ x $是一种歧管时，并且在cotangent捆绑包上给出了非阳性的哈密顿同位素。如果$ x $是具有严格阳性凸度半径的完整的Riemannian歧管，我们证明它是一个很好的度量空间，并且对角的两个双重厚的内核，一个与距离相关，另一个与大地测量流有关。
Given a topological space $X$, a thickening kernel is a monoidal presheaf on $(\mathbb{R}_{\geq0},+)$ with values in the monoidal category of derived kernels on $X$. A bi-thickening kernel is defined on $(\mathbb{R},+)$. To such a thickening kernel, one naturally associates an interleaving distance on the derived category of sheaves on $X$. We prove that a thickening kernel exists and is unique as soon as it is defined on an interval containing $0$, allowing us to construct (bi-)thickenings in two different situations. First, when $X$ is a ``good'' metric space, starting with small usual thickenings of the diagonal. The associated interleaving distance satisfies the stability property and Lipschitz kernels give rise to Lipschitz maps. Second, by using [GKS12], when $X$ is a manifold and one is given a non-positive Hamiltonian isotopy on the cotangent bundle. In case $X$ is a complete Riemannian manifold having a strictly positive convexity radius, we prove that it is a good metric space and that the two bi-thickening kernels of the diagonal, one associated with the distance, the other with the geodesic flow, coincide.
