Factorization Homology

• Factorization Homology is a framework that integrates local algebraic or categorical data over manifolds to produce global homotopical or algebraic invariants.
• It employs a local-to-global approach with excision properties analogous to the Mayer–Vietoris principle, facilitating the treatment of smooth, stratified, and singular spaces.
• It recovers classical invariants such as Hochschild and intersection homology, and extends to applications in topological quantum field theory and nonabelian duality.

Factorization homology, also referred to as topological chiral homology, is a general framework for constructing invariants of manifolds and spaces by "integrating" local algebraic or categorical data over the manifold, with particular efficacy in handling singular, stratified, or structured spaces. The theory merges techniques from manifold topology, higher categories, and algebraic or operadic structures to unify and generalize classical invariants (e.g., singular and Hochschild homology) and to model extended topological field theories and duality phenomena. Developed through foundational works such as (Ayala et al., 2012, Ayala et al., 2014), and (Ayala et al., 2012), factorization homology associates to every n-manifold (possibly singular or stratified) and suitable "disk algebra" a homotopical or algebraic invariant with robust excision, locality, and characterization properties.

1. Local-to-Global Principle and Disk Algebras

Factorization homology is constructed from a local-to-global paradigm. Let $B$ denote a category of basics encoding local singularity or tangential structures (e.g., disks, cones on links, neighborhoods of submanifolds). A Disk$(B)$-algebra $A$ in a symmetric monoidal target category $\mathcal{V}$ (such as chain complexes or spectra) assigns to each basic open $U$ an object $A(U)$ in $\mathcal{V}$. For a $B$-manifold $X$—that is, a space locally modeled on objects of $B$—the factorization homology of $X$ with coefficients in $A$ is defined as

$$J_X(A) \simeq \operatorname{colim}_{U \rightarrow X} A(U)$$

where the colimit is taken over the category of basic opens $U \to X$ (more precisely, the over-category $\operatorname{Disk}(B)/X$) (Ayala et al., 2012). This is an operadic left Kan extension from Disk$(B)$ to the category of $B$-manifolds, extending the assignment $A(-)$ from local models to global objects.

This construction generalizes to structured manifolds (e.g., those equipped with framings, spin, or orientation, as captured by maps $B \to BO(n)$) and to stratified or singular settings. For framed $n$-manifolds, $n$-disk algebras coincide with $E_n$-algebras; in the stratified context, the local models include e.g. cones, links, or products thereof.

2. Excision, Push-Forward, and Characterization Theorems

A defining feature of factorization homology is the excision property: for a collar-gluing $X = X_- \cup_{RV} X_+$ (with $RV$ a collar), there is a canonical equivalence

$$J_X(A) \simeq J_{X_-}(A) \otimes_{J_{RV}(A)} J_{X_+}(A)$$

(Ayala et al., 2012). This is a symmetric monoidal adaptation of the classical Mayer–Vietoris principle, ensuring that global invariants are determined via "gluing" local invariants along overlaps.

Additionally, a push-forward property holds: for a fiber bundle $F$ with fibers $B$-manifolds ($F \to X$), the factorization homology "pushes forward" along $F$, establishing an equivalence between the homology over the base (with coefficients pushed forward) and over the total space.

The central characterization theorem (Ayala et al., 2014) states that homology theories for $B$-manifolds—i.e., symmetric monoidal functors satisfying excision and continuity axioms—are precisely those coming from factorization homology with coefficients in Disk$(B)$-algebras: $$\int : \operatorname{Alg}_{\operatorname{Disk}(B)}(\mathcal{V}) \xrightarrow{\sim} H(\operatorname{Mfld}(B), \mathcal{V})$$ This "universality" parallels axiomatic frameworks like Eilenberg–Steenrod but is formulated in the non-linear (symmetric monoidal and stratified) regime.

The defining axioms are:

1. ⊗–Excision: $H(W_-) \otimes_{H(W_0)} H(W_+) \to H(W)$ is an equivalence for any collar gluing $W = W_- \cup_{W_0} W_+$.
2. Continuity: For an increasing union $X = \bigcup_i W_i$ of open submanifolds, the canonical map $\operatorname{colim}_i H(W_i) \to H(X)$ is an equivalence (Ayala et al., 2014).

3. Connections to Classical Invariants and Topological Quantum Field Theory

Factorization homology recovers and generalizes familiar theories:

• For framed 1-manifolds and associative algebras $A$, one recovers Hochschild homology: $\int_{S^1} A \simeq HH_*(A)$ (Ayala et al., 2012, Ayala et al., 2012).
• For suitable choices of basics (e.g., including stratifications or singularities) and functorial coefficients, intersection homology in the sense of Goresky–MacPherson is obtained.
• When $A$ is a commutative algebra, factorization homology coincides with ordinary homology: $\int_M A \simeq M \otimes A$, and is homotopy invariant.

In topological quantum field theory (TQFT), factorization homology provides a method to "integrate" the algebra of local observables (typically specified by an $n$-disk algebra) over a manifold to produce global state spaces or observable algebras. Observables are local in nature but are coherently glued into global data via the Kan extension. Excision corresponds to the essential "local-to-global" axiom in field theory (Ayala et al., 2012).

4. Nonabelian Poincaré Duality and Sections of Cosheaves

A significant application is the nonabelian generalization of Poincaré duality, established through factorization homology (Ayala et al., 2012, Ayala et al., 2014). When one takes the coefficients to be "group-like" (e.g., $\Omega^n X$ as a Disk$_n$-algebra), there is an equivalence: $\int_X A_E \simeq \Gamma^c_E(X)$ where $A_E$ is the Disk$(B)$-algebra constructed from a connective coefficient system $E$ (a pointed presheaf on $B$), and $\Gamma^c_E(X)$ denotes the space of compactly supported sections of the stratified bundle determined by $E$ over $X$ [(Ayala et al., 2014), Theorem 3.18].

In the smooth or framed context, this specializes to the classical Poincaré duality between homology and cohomology with local coefficients. In stratified or singular situations, intersection homology is one instance where this duality is realized operadically via factorization homology.

5. Stratified Spaces, Singularities, and Link Homology

Factorization homology extends beyond smooth manifolds to categories of stratified or singular manifolds, where the local models are more intricate (e.g., cones or submanifolds modeling singularities) (Ayala et al., 2012, Ayala et al., 2014). In these settings, much of the structure is controlled by the chosen basics and the associated tangential data.

Of special interest is the case of 3-manifolds with properly embedded 1-dimensional submanifolds (knots or links). Here, the relevant algebras are adapted—e.g., Disk$_{3,1}$-algebras—and the theory yields new link invariants. The factorization homology of such pairs (link $L$ inside $M$) with coefficients in a free algebra is computed as

$$\int_{(L \subset M)} \operatorname{Free}(P, Q) \simeq \operatorname{Conf}_P(M \setminus L) \otimes \operatorname{Conf}_Q(L)$$

with $\operatorname{Conf}_P$ and $\operatorname{Conf}_Q$ the configuration space functors associated to label sets $P$, $Q$. This identifies knot and link homology theories, capturing distinctions in the homotopy type of link complements, with precise algebraic input [(Ayala et al., 2014), Proposition 4.13]. These theories are reminiscent of Khovanov homology but arise by operadic and factorization techniques.

6. Algebraic Characterization, Extensions, and Future Directions

An important simplification in the presence of singularities with properly embedded submanifolds is that the homology theories become algebraically tractable. For example, when the submanifold is 1-dimensional inside a 3-manifold, the relevant structures reduce to explicit data involving Disk$_3$- and Disk$_1$-algebras and a compatibility map, giving a higher-categorical generalization of the Deligne conjecture.

The universal characterization theorem implies that, for finitary categories of $B$-manifolds (i.e., built from basics via gluings and unions), all homology theories are determined by their values on basics—thus on the initial "disk-level" data [(Ayala et al., 2014), Corollaries 2.36, 2.40].

A plausible implication is that further extensions to more complicated kinds of singularities or local structures are accessible via this operadic strategy. Indeed, the formalism is amenable to categorical and derived enhancements, as needed in extended TQFTs, string topology, or derived algebraic geometry.

7. References to Key Results and Significance

• The central methodology and characterization theorems are developed in (Ayala et al., 2012, Ayala et al., 2014), which generalize factorization homology beyond smooth manifolds to arbitrary singularities and stratifications.
• Nonabelian Poincaré duality and its stratified versions were established in (Ayala et al., 2012, Ayala et al., 2014) (e.g., Theorem 3.18), and leverage the notion of connective coefficient systems and section spaces.
• The explicit computation for links and knot invariants via configuration spaces and Disk$_{n,k}$-algebras is detailed in [(Ayala et al., 2014), Section 4.3].
• The equivalence between symmetric monoidal homology theories and factorization homology with Disk$(B)$-algebra coefficients is given in [(Ayala et al., 2014), Theorem 2.43].

This synthesis demonstrates that factorization homology provides a robust and axiomatic framework unifying local and global invariants, operadic algebra, and stratified topology. It generalizes the Eilenberg–Steenrod paradigm, enables extension to complex singular geometries, and leads to new invariants in low-dimensional topology, such as knot and link homologies, with deeper implications for duality and field theory.