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• W1637526330 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. As such, 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: $mathcal{B}{sf un}$, an $infty$-category classifying constructible bundles; and $mathcal{E}{sf xit}$, 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." @default.
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• W1637526330 date "2015-02-05" @default.
• W1637526330 modified "2023-10-16" @default.
• W1637526330 title "A stratified homotopy hypothesis" @default.
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