## Abstract

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 language | English (US) |
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Pages (from-to) | 1042-1177 |

Number of pages | 136 |

Journal | Advances in Mathematics |

Volume | 333 |

DOIs | |

State | Published - Jul 31 2018 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)