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.


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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) \,.

This inverse sends an nn-disk algebra

A:Disk nMod k A : Disk_n \to Mod_k

to the functor that sends a manifold XX to the

The factorization homology is then 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 in physics:

Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation


The definition appears in section 3 of

A detailed account is in

A survey that also covers factorization algebras is

See also

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

Last revised on May 14, 2018 at 11:41:55. See the history of this page for a list of all contributions to it.