Fundamental groups of compact Hausdorff spaces

Theory of Compact Hausdorff Shape

In this paper, we aim to establish a new shape theory, compact Hausdorff shape (CH-shape) for general Hausdorff spaces. We use the "internal" method and direct system approach on the homotopy category of compact Hausdorff spaces. Such a construction can preserve most good properties of H-shape given by Rubin and Sanders. Most importantly, we can moreover develop the entire homology theory for CH-shape, including the exactness, dual to the consequence of Mardešić and Segal.

Descent for sheaves on compact Hausdorff spaces

These notes explain some descent results for $\infty$-categories of sheaves on compact Hausdorff spaces and derive some consequences. Specifically, given a compactly assembled $\infty$-category $\mathcal{E}$, we show that the functor sending a locally compact Hausdorff space $X$ to the $\infty$-category $\operatorname{Sh}^{\operatorname{post}}(X;\mathcal{E})$ of Postnikov complete $\mathcal{E}$-valued sheaves on $ X $ satisfies descent for proper surjections. This implies proper descent for left complete derived $\infty$-categories and that the functor $\operatorname{Sh}^{\operatorname{post}}(…

局部紧 Hausdorff 空间上的连续六函子形式体系

Continuous six-functor formalism on locally compact Hausdorff spaces

本文证明了将局部紧 Hausdorff 空间映射到谱层 $\infty$-范畴的函子是连续六函子形式体系的初始对象，并将 Efimov 的代数 $K$-理论计算推广至所有局部化不变量。

The fundamental groupoid of the quotient of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of the induced action

The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space which admits a universal cover is the orbit groupoid of the fundamental groupoid of the space. We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space. The main result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology (Ellis …