TY - JOUR

T1 - Factorization homology I

T2 - Higher categories

AU - Ayala, David

AU - Francis, John

AU - Rozenblyum, Nick

N1 - Funding Information:
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.
Publisher Copyright:
© 2018

PY - 2018/7/31

Y1 - 2018/7/31

N2 - 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.

AB - 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.

KW - (∞,n)-Categories

KW - Exit-path categories

KW - Factorization homology

KW - Stratified spaces

KW - Striation sheaves

KW - Vari-framed stratified manifolds

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U2 - 10.1016/j.aim.2018.05.031

DO - 10.1016/j.aim.2018.05.031

M3 - Article

AN - SCOPUS:85048712054

SN - 0001-8708

VL - 333

SP - 1042

EP - 1177

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -