Cubical Acyclic Homotopy Excision

有向拓扑的立方逼近 I

Cubical approximation for directed topology I

本文研究有向拓扑中的立方逼近理论，证明了立方集和有向空间的几何实现诱导等价，并探讨了有向映射同伦关系的相等条件。

超图同调理论的立方体方法

A cubical approach to homology theories for hypergraphs

本文提出超图的三种同调理论（Γ、□、×），证明其互异且不同于嵌入同调，并引入扩展图离散同伦的超图同伦，发现仅□-同调具同伦不变性，且Γ-同调有独特结构行为。

Homotopy theory of G-diagrams and equivariant excision

Let $G$ be a finite group acting on a small category $I$. We study functors $X \colon I \to \mathscr{C}$ equipped with families of compatible natural transformations that give a kind of generalized $G$-action on $X$. Such objects are called $G$-diagrams. When $\mathscr{C}$ is a sufficiently nice model category we define a model structure on the category of $G$-diagrams in $\mathscr{C}$. There are natural $G$-actions on Bousfield-Kan style homotopy limits and colimits of $G$-diagrams. We prove that weak equivalences between point-wise (co)fibrant $G$-diagrams induce weak $G$-equivalences on hom…

Eilenberg-Steenrod homology and cohomology theories for Čech's closure spaces

We generalize some of the fundamental results of algebraic topology from topological spaces to Čech's closure spaces, also known as pretopological spaces. Using simplicial sets and cubical sets with connections, we define three distinct singular (relative) simplicial and six distinct singular (relative) cubical (co)homology groups of closure spaces. Using acyclic models we show that the three simplicial groups have isomorphic cubical analogues among the six cubical groups. Thus, we obtain a total of six distinct singular (co)homology groups of closure spaces. Each of these is shown to have a c…