Factorization homology I: Higher categories

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 (co, n)-categories. These (co, n)-categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the factorization homology integral(e)(M) of a framed n-manifold M with coefficients in an (co, 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 conitally smooth diffeomorphisms which preserve a vaH-framing is discrete. (C) 2018 Published by Elsevier Inc.