nLab factorization homology


Currently this entry consists mainly of notes taken live in a talk by John Francis at ESI Program on K-Theory and Quantum Fields (2012), without as yet, any double-checking or polishing. So handle with care for the moment.


Higher algebra


algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



Factorization homology is a notion of homology theory for framed nn-dimensional manifolds with coefficients in En-algebras, due to (Francis). It is similar in spirit to factorization algebras, blob homology and topological chiral homology. In fact the definition of factorization homology turns out to be equivalent to that of topological chiral topology (Francis b).


Write Mfd n Mfd_n^{\coprod} for the category of manifolds with embeddings as morphisms. This is naturally a topological category, hence regard it as an (infinity,1)-category. Regard it furthermore as a symmetric monoidal (∞,1)-category with tensor product given by disjoint union.

For kk a field, write Mod kMod_k for the symmetric monoidal (∞,1)-category of kk-chain complexes.

Let H(Mfd n ,Mod k)H(Mfd_n^{\coprod}, Mod_k) be the sub-(∞,1)-category of those monoidal (∞,1)-functors F:Mfd n opMod kF : Mfd_n^{op} \to Mod_k which are “cosheaves” in that for any decomposition of a manifold XX into submanifolds XX' and XX'' with overlap OO, we have an equivalence

F(X)F(X) F(O)F(X). F(X) \simeq F(X') \otimes_{F(O)}F(X'') \,.

Next, let Disk nMfd nDisk_n \subset Mfd_n be the full sub-(∞,1)-category on those manifolds which are finite disjoint unions of the Cartesian space n\mathbb{R}^n.

Restriction along this inclusion gives an (∞,1)-functor

H(Mfd n,Mod k)Disk nAlg(Mod k) H(Mfd_n, Mod_k) \to Disk_n-Alg (Mod_k)

This turns out to be an equivalence of (∞,1)-categories. The inverse is defined to be factorization homology

FactorizationHomology:Disk nAlg(Mod k)H(Mfd n,Mod k). FactorizationHomology : Disk_n-Alg (Mod_k) \to H(Mfd_n, Mod_k) \,.

which sends an nn-disk algebra A:Disk nMod kA : Disk_n \to Mod_k to the functor that sends a manifold XX to the derived coend

XA=𝔼 X Disk nA \int^X A = \mathbb{E}_X \otimes_{Disk_n} A

of AA with

𝔼 X:Disk nEmb(,X)TopC ()Mod k. \mathbb{E}_X : Disk_n \stackrel{Emb(-,X)}{\to} Top \stackrel{C_\bullet(-)}{\to} Mod_k \,.

This is equivalent to topological chiral homology, to be thought of as a topological version of chiral algebras. A version with values in homotopy types instead of chain complexes was given by Salvatore and Graeme Segal.


Relation to cobordism hypothsis

From a functor FH(Mfd n,Mod k)F \in H(Mfd_n, Mod_k) we get an extended TQFT with values in kk-linear (,n)(\infty,n)-categories

Z F:Bord nCat n(k)Z_F : Bord_n \to Cat_n(k) which sends a kk-manifold XX to F(X× nk)F(X \times \mathbb{R}^{n-k}), regarded as a bimodule between the analogous boundary restriction, and hence as a k-morphism in Cat n(k)Cat_n(k).

From a Disk nDisk_n-algebra AA we obtain the corresponding delooping BA(Cat n(k) dualizable) O(n)\mathbf{B}A \in (Cat_n(k)_{dualizable})^{O(n)} which is a kk-linear (infinity,n)-category that is a fully dualizable object. The cobordism hypothesis identifies this with cobordism representations, and the claim is that this identification is compatible factorization homology.


Dimension 1

A Disk 1Disk_1-algebra AA in Mod kMod_k is equivalently a differential graded algebra.

The value of the corresponding F AH(Mfd 1,Mod k)F_A \in H(Mfd_1, Mod_k) on the circle is the Hochschild homology of AA

S 1A 1A S 0×A 1AHH (A). \int_{S^1} A \simeq \int_{\mathbb{R}^1} A \otimes_{\int_{S^0 \times \mathbb{R}}A} \int_{\mathbb{R}^1} A \simeq HH_\bullet(A) \,.

From nn-fold loop spaces

Given a topological space ZZ we get a Disk nDisk_n-algebra

Disk n Maps compact(,Z)TopC *()Mod k Disk_n^\coprod \stackrel{Maps_{compact}(-,Z)}{\to} Top \stackrel{C_\ast(-)}{\to} Mod_k

Where Maps compact( n,Z)Ω nZMaps_{compact}(\mathbb{R}^n, Z) \simeq \Omega^n Z is the n-fold loop space of ZZ.

Theorem (Salvatore and Lurie)

If ZZ is (n1)(n-1)-n-connected object of an (infinity,1)-category

XC *(Ω nZ)C *Maps compact(X,Z). \int_X C_\ast(\Omega^n Z) \simeq C_\ast Maps_{compact}(X,Z) \,.

duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}



The definition appears in section 3 of

A detailed account is in

A survey that also covers factorization algebras is

See also

Generalization to orbifolds:

Some applications are

Application to higher Hochschild cohomology is discussed in

Application to stratified spaces with tangential structures is discussed in

A duality theorem for factorization homology, generalizing Poincare duality for manifolds and Koszul duality for E-n algebras.

Discussion in the context of extended TQFT appears in

For surfaces equipped with flat connections for a finite group:


Relation to cohomology of configuration spaces

Expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces, and relation to representation stability:

  • Quoc P. Ho, Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras [arXiv:2004.00252]

Last revised on December 13, 2023 at 18:18:46. See the history of this page for a list of all contributions to it.