论文标题
方便的本地分层空间
A convenient category of locally stratified spaces
论文作者
论文摘要
在本文中，我们定义了局部分层空间的概念。局部分层的空间是在分层空间上局部建模的特定类型的流和D空间。我们构建了本地分层空间的本地和笛卡尔封闭类别，该类别接纳了与简单集的类别相邻的。此外，我们表明，由本地分层的空格跨越的完整子类别，其相关的简单集是QuasicateGory具有带有纤毛对象的类别的结构。我们定义了本地分层空间的基本类别，并表明从简单集的基本类别到其实现的基本类别的规范函数本质上是溢出的。我们表明，这样的函子将分裂的单态发送给同构，特别是我们表明，它不一定是类别的等效性。另一方面，我们表明，实现圆圈的基本类别等同于自然数的单体。总而言之，我们定义了本地分层空间的左侧封面，并表明，在适当的假设下，简单集的基本类别的表示类别等于其实现的左覆盖类别。
In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian closed category of locally stratified spaces that admits an adjunction with the category of simplicial sets. Moreover, we show that the full subcategory spanned by locally stratified spaces whose associated simplicial set is a quasicategory has the structure of a category with fibrant objects. We define the fundamental category of a locally stratified space and show that the canonical functor from the fundamental category of a simplicial set to the fundamental category of its realisation is essentially surjective. We show that such a functor sends split monomorphisms to isomorphisms, in particular we show that it is not necessarily an equivalence of categories. On the other hand, we show that the fundamental category of the realisation of the simplicial circle is equivalent to the monoid of the natural numbers. To conclude, we define left covers of locally stratified spaces and we show that, under suitable assumptions, the category of representations of the fundamental category of a simplicial set is equivalent to the category of left covers over its realisation.
