论文标题
交叉共同体中的自然操作
Natural operations in Intersection Cohomology
论文作者
论文摘要
Eilenberg-Maclane空间（将拓扑空间的奇异共同体学群体分类）在简单集合的框架中接受了自然结构。 M. Goresky和R. Macpherson的长期开放问题是，在分层空间的交点共同体组中存在相似的空间。这项工作的一个功能是构造这样的简单组合。从R. Macpherson，J。Lurie等人的作品中，现在通常可以接受的是，与拓扑空间相关的简单简单集必须被尊重分层的单数简单简单组所取代。这是在地层poset神经上的简单集类别中编码的。对于每种变态，我们从其定义一个函子，在通勤环上的科链复合物类别中的值。该结构基于简单的爆炸，相关的共同点是M. Goresky和R. Macpherson定义的交叉点协同学。该函子接收一个伴随，我们使用它来获得交叉共同体的分类空间。从这些分类空间的交集共同体学来理解自然交叉的共同体学操作。与经典的情况一样，它们形成无限的环路空间。在最后一部分中，我们检查了一个只有一个单数层的分层空间的深度。我们观察到分类空间是乔伊尔在经典的Eilenberg-Maclane空间上的投射锥。我们建立了他们的一些特性，并猜想，对于戈尔斯基和麦克弗森的变态，所有相交的共同体操作均由经典诱导。
Eilenberg-MacLane spaces, that classify the singular cohomology groups of topological spaces, admit natural constructions in the framework of simplicial sets. The existence of similar spaces for the intersection cohomology groups of a stratified space is a long-standing open problem asked by M. Goresky and R. MacPherson. One feature of this work is a construction of such simplicial sets. From works of R. MacPherson, J. Lurie and others, it is now commonly accepted that the simplicial set of singular simplices associated to a topological space has to be replaced by the simplicial set of singular simplices that respect the stratification. This is encoded in the category of simplicial sets over the nerve of the poset of strata. For each perversity, we define a functor from it, with values in the category of cochain complexes over a commutative ring. This construction is based upon a simplicial blow up and the associated cohomology is the intersection cohomology as it was defined by M. Goresky and R. MacPherson. This functor admits an adjoint and we use it to get classifying spaces for intersection cohomology. Natural intersection cohomology operations are understood in terms of intersection cohomology of these classifying spaces. As in the classical case, they form infinite loop spaces. In the last section, we examine the depth one case of stratified spaces with only one singular stratum. We observe that the classifying spaces are Joyal's projective cones over classical Eilenberg-MacLane spaces. We establish some of their properties and conjecture that, for Goresky and MacPherson perversities, all intersection cohomology operations are induced by classical ones.
