## 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 |

### Funding

DA was supported by the National Science Foundation under award 1507704. JF was supported by the National Science Foundation under awards 1207758 and 1508040. Parts of this paper were written while JF was a visitor at the Mathematical Sciences Research Institute and at Université Pierre et Marie Curie.

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics