Wiki. “平展同伦论” [平展同伦论]

平展同伦的概念可理解为如下经典事实的推广: 考虑仿紧 (paracompact) 拓扑空间 $X$ 的好覆盖 (即任意有限个开集的交为空或可缩) $\{U_i \to X\}$ 的 Cech 脉, 逐项取其连通分支的集合, 所得的单纯集表现了 $X$ 的同伦型.

更一般地, 对于局部可缩空间 $X$, 没有好覆盖的概念, 但可以考虑超覆盖的类似构造, 对所有超覆盖给出的单纯集取极限.

在任何景上都可以做类似的事情. 在平展景上做的事情叫做平展同伦论.

But the concept is much more general. In particular, one can understand the construction of the limit over contractions of hypercovers as a presentation of naturally defined (∞,1)-functors in (∞,1)-topos theory.

Notably, if the given site is a locally ∞-connected site, then the étale homotopy construction computes precisely the derived functor that presents the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos. Many constructions in the literature can be understood as being explicit realizations of this simple general concept.

相关概念

∞-意象的形