链环空间与出口路径
Linked spaces and exit paths
AI总结：
本文将深度为1偏序集上的锥形光滑空间分层拓扑归约为链环光滑流形的普通拓扑，引入显式出口路径拟范畴并证明其诱导完全忠实函子，同时否定解决了若干较弱版本的猜想。
摘要：
由Ayala、Francis和Tanaka引入的锥形光滑空间（CSSs）构成了一大类包含Whitney分层空间的奇异空间。我们将深度为1的偏序集上的CSS的分层拓扑归约为链环光滑流形的普通拓扑，即光滑流形的span $M\xleftarrowπ L\xrightarrowιN$，其中 $π$ 是纤维丛，$ι$ 是闭嵌入。为此，我们为链环空间引入了显式的出口路径拟范畴（EPCs），并证明这诱导了一个从链环空间的拟范畴到所有拟范畴的拟范畴的完全忠实函子，其本质像包含深度为1的偏序集上的CSS的Lurie--MacPherson EPCs。我们利用链环光滑流形，通过展示具有到 $\{0<1\}$ 的保守函子且满足某些有限性条件但不等价于CSS的EPCs的拟范畴，以否定方式解决了Ayala--Francis--Rozenblyum猜想的各种较弱版本。在续篇中，我们为链环光滑流形发展了切向理论，并将深度为1的偏序集上的锥形光滑丛的分类归约为链环光滑流形上的普通丛的分类。
英文摘要：
Conically smooth spaces (CSSs), introduced by Ayala, Francis and Tanaka, constitute a large class of singular spaces including Whitney-stratified spaces. We reduce the stratified topology of CSSs over depth-$1$ posets to the ordinary topology of linked smooth manifolds, i.e., spans $M\xleftarrowπ L\xrightarrowιN$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. To that end, we introduce explicit exit path quasi-categories (EPCs) for linked spaces and prove that this induces a fully faithful functor from a quasi-category of linked spaces to the quasi-category of all quasi-categories whose essential image includes the Lurie--MacPherson EPCs of CSSs over depth-$1$ posets. We use linked smooth manifolds to resolve various weaker versions of a conjecture of Ayala--Francis--Rozenblyum in the negative by exhibiting quasi-categories with conservative functors to $\{0<1\}$ satisfying certain finiteness conditions but which are not equivalent to EPCs of CSSs. In a sequel, we develop a tangential theory for linked smooth manifolds and reduce the classification conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds.
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