In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. We then define a generalization of the homotopy groups for any fibrant filtered simplicial set $X$ : the filtered homotopy groups $s\pi_n(X)$. They are filtered simplicial sets built from the homotopy groups of the different pieces of $X$. We then show that the weak equivalences are exactly the morphisms that induce isomorphisms on those filtered homotopy groups. Then, using filtered versions of the topological realisation of a simplicial set and of the simplicial set of singular simplices, we transfer those results to a category whose objects are topological spaces stratified over $P$. In particular, we get a stratified version of Whitehead's theorem. Specializing to the case of conically stratified spaces, a wide class of topological stratified spaces, we recover a theorem of Miller saying that to understand the homotopy type of conically stratified spaces, one only has to understand the homotopy type of strata and holinks. We then provide a family of examples of conically stratified spaces and of computations of their filtered homotopy groups.

中文翻译：

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分层同伦理论的简单方法

在本文中，我们从简单的角度考虑分层空间的同伦理论。我们首先考虑某个固定偏序$P$上的过滤单纯集的模型类别，并证明它是一个单纯组合模型类别。然后，我们为任何 fibrant 过滤单纯集 $X$ 定义同伦群的泛化：过滤同伦群 $s\pi_n(X)$。它们是从 $X$ 的不同部分的同伦群构建的过滤单纯集。然后我们证明弱等价正是在那些过滤的同伦群上诱导同构的态射。然后，使用单纯集的拓扑实现和奇异单纯形的单纯集的过滤版本，我们将这些结果转移到一个类别，该类别的对象是在 $P$ 上分层的拓扑空间。特别是，我们得到了怀特海定理的分层版本。专门研究圆锥分层空间的情况，一大类拓扑分层空间，我们恢复了米勒的一个定理，他说要理解圆锥分层空间的同伦类型，只需要理解层和全链接的同伦类型。然后，我们提供了一系列圆锥分层空间的示例及其过滤同伦群的计算。