JournalsjemsVol. 21, No. 4pp. 1071–1178

A stratified homotopy hypothesis

  • David Ayala

    Montana State University, Bozeman, USA
  • John Francis

    Northwestern University, Evanston, USA
  • Nick Rozenblyum

    University of Chicago, USA
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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 R1\mathbb R^1-invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in A1\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: Bun\mathcal B\mathsf{un}, an \infty-category classifying constructible bundles; and Exit\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.

Cite this article

David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis. J. Eur. Math. Soc. 21 (2019), no. 4, pp. 1071–1178

DOI 10.4171/JEMS/856