The stratified Grassmannian and its depth-one subcategories
Ödül Tetik
TL;DR
The paper develops a hands-on tangential framework for depth-1 linked smooth spans, replacing AFR’s stratified tangent classifier with a concrete model in terms of the stratified Grassmannian $\mathbf{V}^{\hookrightarrow}$ and the unpacking functor $\mathbf{U}: \mathbf{EX}(B\mathrm{O}(n,m))\to\mathbf{V}^{\hookrightarrow}$. It proves that $\mathbf{U}$ is fully faithful, identifying $\mathbf{EX}(B\mathrm{O}(n,m))$ with the depth-1 subcategory $\mathbf{V}^{\hookrightarrow}|_{n,n+m}$, and shows how tangential structures on depth-1 CSSs pull back along this embedding. Consequently, moduli of conically smooth vector and principal bundles over depth-1 CSSs reduce to classical bundle data on linked manifolds, with corollaries for variframes and Stiefel-type stratifications. The construction provides explicit, inductive control of higher morphisms via the exit-path formalism, yielding a practical bridge between stratified tangent theory and ordinary bundle classification in depth-1. Overall, the work offers a concrete, computationally tractable pathway to classify stratified tangential structures by reducing to ordinary bundles on linked manifolds.
Abstract
We introduce a tangential theory for linked smooth 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 $\mathbf{EX}$, introduced in a prequel, 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. We thus reduce the classification of conically smooth bundles over depth-$1$ posets to that of ordinary bundles on linked smooth manifolds.