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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We show that conically smooth stratified spaces embed fully faithfully into -categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. Hence, each -category defines a stack on conically smooth stratified spaces, and we identify the descent conditions it satisfies. These include -invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in -homotopy theory. In this way, we identify -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 -categories: , an -category classifying constructible bundles; and , the absolute exit-path -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.

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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