Cellular Stratified Spaces II: Basic Constructions

Cellular stratified spaces

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonz{á}lez, and Rudyak with the aim of constructing a cellular model of the configuration space of a sphere. Although the original aim was not achieved in the project, the notion of cellular stratified spaces turns out to be useful, at least, in the study of configuration spaces of graphs. In particular, the notion of totally normal cellular stratified spaces was used successfully in a joint work with the former students of the author arXiv:1312.7368 to study the homotopy type of configuration sp…

Totally normal cellular stratified spaces and applications to the configuration space of graphs

The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced by the third author by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's th…

Cellular Stratified Spaces I: Face Categories and Classifying Spaces

The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally normal regular cellular stratified space $X$ can be embedded in $X$ as a strong deformation retract. Here we elaborate on this idea and develop the theory of cellular stratified spaces. We introduce the notion of cylindrically normal cellular stratified spaces and associate a top…

拓扑∞-栈的分层同伦理论：工具箱

Stratified homotopy theory of topological $\infty$-stacks: a toolbox

本研究借助∞-栈理论，为分层拓扑栈的分层同伦理论提供基础定义与计算工具。

Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory

This paper develops the foundations of a simplicial theory of weak omega-categories, which builds upon the insights originally expounded by Ross Street in his 1987 paper on oriented simplices. The resulting theory of weak complicial sets provides a common generalisation of the theories of (strict) omega-categories, Kan complexes and Joyal's quasi-categories. We generalise a number of results due to the current author with regard to complicial sets and strict omega-categories to provide an armoury of well behaved technical devices, such as joins and Gray tensor products, which will be used to s…