The stratified Grassmannian and its depth-one subcategories

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.