Factorization homology I: Higher categories

David Ayala, John Francis, Nick Rozenblyum

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together with a coherent system of compatibilities of framings along links between strata. Our main result constructs labeling systems on disk-stratified vari-framed n-manifolds from (∞,n)-categories. These (∞,n)-categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the factorization homology ∫MC of a framed n-manifold M with coefficients in an (∞,n)-category C is the classifying space of C-labeled disk-stratifications over M. The core calculation underlying our main result is the following: for any disk-stratified manifold, the space of conically smooth diffeomorphisms which preserve a vari-framing is discrete.

Original languageEnglish (US)
Pages (from-to)1042-1177
Number of pages136
JournalAdvances in Mathematics
StatePublished - Jul 31 2018


  • (∞,n)-Categories
  • Exit-path categories
  • Factorization homology
  • Stratified spaces
  • Striation sheaves
  • Vari-framed stratified manifolds

ASJC Scopus subject areas

  • Mathematics(all)


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