A stratified homotopy hypothesis

Abstract

We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. Hence, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions it satisfies. These include ${\mathbb{R}}^1$-invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in ${\mathbb{A}}^1$-homotopy theory. In this way, we identify $\infty$-categories as striation sheaves, which are those sheaves on conically smooth stratified spaces satisfying the indicated descent. We use this identification to construct by hand two remarkable examples of $\infty$-categories: $\mathcal{B}\mathsf{un}$, an $\infty$-category classifying constructible bundles; and $\mathcal{E}\mathsf{xit}$, the absolute exit-path $\infty$-category. These constructions are deeply premised on stratified geometry, the key geometric input being a characterization of conically smooth stratified maps between cones and the existence of pullbacks for constructible bundles.