论文标题
分层的格拉斯曼尼亚及其深度为一子类别
The stratified Grassmannian and its depth-one subcategories
论文作者
论文摘要
我们为链接的深度$ 1 $（即，对于跨度$ \ mathfrak {s} =（m \oftsetousπ{\ twebtheadleftarrow} l \oversetι{\ hook frightArrow} n）$ flicolds $ fiber和$ fiber的链接。 $ \ mathfrak {s} $的切线分类器作为拓扑跨度映射$ \ Mathfrak {s} \ to B \ Mathrm {o}（n，m）$其中$ b \ mathrm {o {o}（o） B \ Mathrm {O}（M）\ HookrightArrow B \ Mathrm {O}（N+M））$。我们表明，这通过构造完全忠实的函数$ \ mathbf {ex}（b \ m mathrm {o}（o}（n，m）），这将恢复并概括由Ayala，Francis和Rozenblyum介绍的切线平滑分层空间所介绍的切向理论。 $ \ mathbf {ex} $采用跨度的出口路径准类别，$ \ mathbf {v}^{\ hookrightarrow} $是AFR的无限分层法的准类别模型。该结果具有其他经典结构组和stiefel歧管的类似物。作为一个应用程序，我们将圆锥光滑捆绑包的分类减少到链接歧管上的普通捆绑包的分类。
We introduce a tangential theory for linked manifolds of depth $1$, i.e., for spans $\mathfrak{S}=(M\oversetπ{\twoheadleftarrow} L\oversetι{\hookrightarrow}N)$ of smooth manifolds where $π$ is a fibre bundle and $ι$ is a closed embedding. The tangent classifier of $\mathfrak{S}$ is given as a topological span map $\mathfrak{S}\to B\mathrm{O}(n,m)$ where $B\mathrm{O}(n,m)=(B\mathrm{O}(n)\twoheadleftarrow B\mathrm{O}(n)\times B\mathrm{O}(m)\hookrightarrow B\mathrm{O}(n+m))$. We show that this recovers and generalises the tangential theory introduced by Ayala, Francis and Rozenblyum for conically smooth stratified spaces by constructing fully faithful functors $\mathbf{EX}(B\mathrm{O}(n,m))\hookrightarrow\mathbf{V}^{\hookrightarrow}$ of quasi-categories, where $\mathbf{EX}$ takes the exit path quasi-category of the span, and $\mathbf{V}^{\hookrightarrow}$ is a quasi-category model of the infinite stratified Grassmannian of AFR. This result has analogues for other classical structure groups and for Stiefel manifolds. As an application, we reduce the classification of conically smooth bundles in depth $1$ to the classification of ordinary bundles on linked manifolds.
