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Factorization homology I: higher categories

David Ayala, John Francis, Nick Rozenblyum

TL;DR

Factorization homology for framed n-manifolds is developed by pairing stratified geometry with (∞,n)-categories via labeling systems on disk-stratified vari-framed manifolds. The authors build a three-step program: (i) define labeling systems as an ∞-category on disk-stratified vari-framed spaces, (ii) embed the ∞-category of (∞,n)-categories into labeling systems through a cellular realization from Θn, and (iii) extend to general vari-framed manifolds by left Kan extension, yielding ∫M C as the classifying space of C-labeled disk-stratifications. Key innovations include vari-framings (a stratified, coherent framing across links), striation-sheaves equating with ∞-categories, and a cellular realization ⟨-⟩: Θn^op → cDisk^vfr_n, which ground the connection between higher categories and disk-stratified topology. The main result states that there is a functor ∫: Cat_(∞,n) → Fun(cMfd_n^vfr, Spaces) which is fully faithful for n<3, enabling space-valued invariants of smooth framed n-manifolds and paving the way toward fully extended TQFTs in future work. This framework promises a robust bridge between manifold topology and higher category theory, with potential applications to cobordism-type field theories and geometric-topological invariants computed via factorization homology.

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 $(\infty,n)$-categories. These $(\infty,n)$-categories, in contrast with the literature to date, are not required to have adjoints. 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 $\cC$-labeled disk-stratifications over $M$.

Factorization homology I: higher categories

TL;DR

Factorization homology for framed n-manifolds is developed by pairing stratified geometry with (∞,n)-categories via labeling systems on disk-stratified vari-framed manifolds. The authors build a three-step program: (i) define labeling systems as an ∞-category on disk-stratified vari-framed spaces, (ii) embed the ∞-category of (∞,n)-categories into labeling systems through a cellular realization from Θn, and (iii) extend to general vari-framed manifolds by left Kan extension, yielding ∫M C as the classifying space of C-labeled disk-stratifications. Key innovations include vari-framings (a stratified, coherent framing across links), striation-sheaves equating with ∞-categories, and a cellular realization ⟨-⟩: Θn^op → cDisk^vfr_n, which ground the connection between higher categories and disk-stratified topology. The main result states that there is a functor ∫: Cat_(∞,n) → Fun(cMfd_n^vfr, Spaces) which is fully faithful for n<3, enabling space-valued invariants of smooth framed n-manifolds and paving the way toward fully extended TQFTs in future work. This framework promises a robust bridge between manifold topology and higher category theory, with potential applications to cobordism-type field theories and geometric-topological invariants computed via factorization homology.

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 -manifolds from -categories. These -categories, in contrast with the literature to date, are not required to have adjoints. This allows the following conceptual definition: the factorization homology of a framed -manifold with coefficients in an -category is the classifying space of -labeled disk-stratifications over .

Paper Structure

This paper contains 48 sections, 53 theorems, 326 equations, 14 figures.

Table of Contents

  1. Recollections of striation sheaves
  2. Stratified spaces
  3. Exit-paths
  4. Striation sheaves
  5. Classes of morphisms
  6. Tangential structures
  7. Constructible tangent bundle
  8. Synopsis
  9. Vector spaces
  10. The constructible tangent bundle
  11. Elucidating $\mathsf T^{\sf fib}$
  12. Framings: one stratified space at a time
  13. Solid framings
  14. Explicating framings
  15. Characterization of the constructible tangent bundle
  16. ...and 33 more sections

Key Result

Theorem 2

Per the erratum of §sec.erratum, this functor, which is defined in Definition def.fact.homology in §sec.fact.def, need not be fully faithful for $n \geq 3$ as was asserted in the previous version of this work. There is functor from the $\infty$-category of $(\mathop{\mathrm{\infty}}\nolimits,n)$-cat For $n<3$, this functor is fully faithful.

Figures (14)

  • Figure 1: A refinement morphism in $\mathop{\mathrm{\sf c}\mathcal{B}\mathsf{un}}\nolimits$ between two stratifications of the closed interval.
  • Figure 2: A creation morphism in $\mathop{\mathrm{\sf c}\mathcal{B}\mathsf{un}}\nolimits$ between two stratifications of the closed interval.
  • Figure 3: A framed stratified $2$-manifold with its constructible tangent bundle; the 1-dimensional strata are labelled by the embedding of the tangent bundle.
  • Figure 4: A vari-framed hemispherically stratified $1$-disk with strata labelled by the poset.
  • Figure 5: A vari-framed hemispherically stratified $2$-disk with stata labelled by the poset.
  • ...and 9 more figures

Theorems & Definitions (206)

  • Theorem 2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Remark 1.9
  • Definition 1.10: $\mathop{\mathrm{\mathcal{S}\mathsf{trat}}}\nolimits$
  • ...and 196 more