Factorization homology from higher categories

David Ayala, John Francis, Nick Rozenblyum

Published 2015-04-15Version 1

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 of strata. Our main result constructs labeling systems on disk-stratified vari-framed $n$-manifolds from $(\infty,n)$-categories. These $(\infty,n)$-categories, in contrast with the literature to date, are not required to have adjoints. The core calculation supporting this result is a homotopy equivalence between the space of conically smooth diffeomorphisms of a disk-stratified manifold and its space of vari-framings. This allows the following conceptual definition: the factorization homology \[ \int_M\mathcal{C} \] of a framed $n$-manifold $M$ with coefficients in an $(\infty,n)$-category $\mathcal{C}$ is the classifying space of $\mathcal{C}$-labeled disk-stratifications over $M$.