A stratified homotopy hypothesis

David Ayala, John Francis, Nick Rozenblyum

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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 R 1 -invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in A 1 -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: Bun, an ∞-category classifying constructible bundles; and Exit, 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.

Original languageEnglish (US)
Pages (from-to)1071-1178
Number of pages108
JournalJournal of the European Mathematical Society
Issue number4
StatePublished - 2019


  • Blowups
  • Complete Segal spaces
  • Constructible bundles
  • Constructible sheaves
  • Exit-path category
  • Quasi-categories
  • Resolution of singularities
  • Stratified spaces
  • Striation sheaves
  • Transversality sheaves
  • ∞-categories

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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