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A factorization homology primer

David Ayala, John Francis

TL;DR

This primer introduces factorization homology in the topological setting, unifying an algebraic input (n-disk algebras) with a geometric manifold input to produce global invariants ∫_M A in a symmetric monoidal ∞-category. It develops a comprehensive framework: tangential structures and framing via tangent classifiers, Weiss and disk approaches to sheaves, and boundary-aware variants; it proves a pushforward (excision) principle and characterizes homology theories as left Kan extensions from disk algebras. The text then establishes nonabelian Poincaré duality, provides explicit calculations for common inputs (direct sum, commutative, Lie, and free disk algebras), and introduces filtrations (cardinality and Goodwillie) that organize factors by configuration-space data and polynomial approximations. It culminates with Poincaré/Koszul duality, and an outline for extending factorization homology to structured singular manifolds, linking topological quantum field theory, deformation theory, and derived geometry through a coherent, highly structured framework.

Abstract

This chapter amalgamates some foundational developments and calculations in factorization homology.

A factorization homology primer

TL;DR

This primer introduces factorization homology in the topological setting, unifying an algebraic input (n-disk algebras) with a geometric manifold input to produce global invariants ∫_M A in a symmetric monoidal ∞-category. It develops a comprehensive framework: tangential structures and framing via tangent classifiers, Weiss and disk approaches to sheaves, and boundary-aware variants; it proves a pushforward (excision) principle and characterizes homology theories as left Kan extensions from disk algebras. The text then establishes nonabelian Poincaré duality, provides explicit calculations for common inputs (direct sum, commutative, Lie, and free disk algebras), and introduces filtrations (cardinality and Goodwillie) that organize factors by configuration-space data and polynomial approximations. It culminates with Poincaré/Koszul duality, and an outline for extending factorization homology to structured singular manifolds, linking topological quantum field theory, deformation theory, and derived geometry through a coherent, highly structured framework.

Abstract

This chapter amalgamates some foundational developments and calculations in factorization homology.

Paper Structure

This paper contains 32 sections, 49 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Outline of contents
  3. Manifolds with tangential structure
  4. Manifolds and embeddings
  5. Sheaves on $n$-manifolds
  6. Tangent classifier
  7. $B$-framed manifolds
  8. Weiss sheaves on $n$-manifolds
  9. Disks
  10. Manifolds with boundary
  11. Localizing with respect to isotopy equivalences
  12. Homology theories for manifolds
  13. Disk algebras
  14. Factorization algebras
  15. Factorization homology over oriented $1$-manifolds with boundary
  16. ...and 17 more sections

Key Result

Proposition 2.6

The continuous homomorphism of topological monoids $\mathsf O(n) \to \mathop{\mathrm{\mathsf{Emb}}}\nolimits(\mathbb R^n,\mathbb R^n)$ is a homotopy equivalence.

Theorems & Definitions (163)

  • Definition 2.1
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10: aft1
  • Remark 2.11
  • ...and 153 more