The stratified Grassmannian and its depth-one subcategories

TL;DR

This paper develops a new tangential theory for depth-one linked smooth manifolds, generalizing existing stratified space theories and connecting bundle classification to classical structures.

Contribution

It introduces a tangential theory for depth-one manifolds, generalizes stratified space theory, and relates bundle classification to classical structure groups.

Findings

Constructs fully faithful functors between exit path quasi-categories and stratified Grassmannians.

Generalizes tangential theories to classical structure groups and Stiefel manifolds.

Reduces classification of bundles over depth-one posets to classical bundle theory.

Abstract

We introduce a tangential theory for linked smooth manifolds of depth $1$, i.e., for spans $\mathfrak{S}=(M\overset{\pi}{\twoheadleftarrow} L\overset{\iota}{\hookrightarrow}N)$ of smooth manifolds where $\pi$ is a fibre bundle and $\iota$ 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.