Univalence for inverse EI diagrams

Persistence Diagrams with Linear Machine Learning Models

Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be extracted by applying machine learnings. In particular, the ability for explicitly analyzing the inverse in the original data space from those statistical features of persistence diagrams is significantly important for practical applications. In this paper, we propose a unified method for the inverse analysis by combining linear machine learning models with persis…

From Geometry to Topology: Inverse Theorems for Distributed Persistence

What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic. We therefore propose that the correct invariant is not the persistence diagram of X, but rather the collection of persistence diagrams of many small subsets. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of point clouds (with the qu…

环图上模的同伦理论

Homotopy theory of modules over diagrams of rings

给定一个环的图，我们可以考虑环上的模的范畴。我们对这类范畴的同伦理论感兴趣：给定一个合适的模型范畴M（s）的图（当s贯穿图时），我们考虑在s处的对象X（s）来自M（s）的图的范畴。我们开发的模型结构，这些类别的图，和Quillen解释，相关类别的基础上，不同的图表形状。在一定条件下，这些映射的胞元化（或右Bousfield局部化）诱导Quillen等价。作为一个应用，我们证明了环谱图（或微分分次环）上模范畴的胞元化与环的逆极限上模的Quillen等价.另一个应用程序的一般机械在这里是在工作中的作者对代数模型的理性等变谱。其中一些材料最初出现在预印本“An algebraic model for rational torus-equivariant stable homotopy theory”，arXiv：1101.2511，但在这里被推广。

群图的非阿贝尔同调的同伦余终结性

Homotopy cofinality for Non-Abelian homology of group diagrams

本文证明了同伦余终结函子诱导同伦余极限弱等价，据此确立群图非阿贝尔同调在同伦余终结函子原像及左 Kan 扩张下的不变性，并推广构建了单纯集与预层的非阿贝尔 Gabriel-Zisman 及 Baues-Wirsching 同调理论。