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.

Idea

Factorization homology is a notion of homology theory for framed $n$-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).

Definition

Write ${\mathrm{Mfd}}_n^{\coprod }$ 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 $k$ a field, write ${\mathrm{Mod}}_k$ for the symmetric monoidal (∞,1)-category of $k$-chain complexes.

Let $H({\mathrm{Mfd}}_n^{\coprod },{\mathrm{Mod}}_k)$ be the sub-(∞,1)-category of those monoidal (∞,1)-functors $F:{\mathrm{Mfd}}_n^{\mathrm{op}}\to {\mathrm{Mod}}_k$ which are “cosheaves” in that for any decomposition of a manifold $X$ into submanifolds $X\prime$ and $X″$ with overlap $O$, we have an equivalence

$$F(X)\simeq F(X\prime ){\otimes }_{F(O)F(X″)}.$$F(X) \simeq F(X') \otimes_{F(O) F(X'')} \,.

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

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

$$H({\mathrm{Mfd}}_n,{\mathrm{Mod}}_k)\to {\mathrm{Disk}}_n-\mathrm{Alg}({\mathrm{Mod}}_k)$$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

$$\mathrm{FactorizationHomology}:{\mathrm{Disk}}_n-\mathrm{Alg}({\mathrm{Mod}}_k)\to H({\mathrm{Mfd}}_n,{\mathrm{Mod}}_k).$$FactorizationHomology : Disk_n-Alg (Mod_k) \to H(Mfd_n, Mod_k) \,.

This inverse sends an $n$-disk algebra

$$A:{\mathrm{Disk}}_n\to {\mathrm{Mod}}_k$$A : Disk_n \to Mod_k

to the functor that sends a manifold $X$ to the

The factorization homology is then the derived coend

$${\int }^{X}A={𝔼}_X{\otimes }_{{\mathrm{Disk}}_n}A$$\int^X A = \mathbb{E}_X \otimes_{Disk_n} A

of $A$ with

$${𝔼}_X:{\mathrm{Disk}}_n\stackrel{\mathrm{Emb}(-,X)}{\to }\mathrm{Top}\stackrel{C_{•}(-)}{\to }{\mathrm{Mod}}_k.$$\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.

Properties

Relation to cobordism hypothsis

From a functor $F\in H({\mathrm{Mfd}}_n,{\mathrm{Mod}}_k)$ we get an extended TQFT with values in $k$-linear $(\mathrm{\infty },n)$-categories

$Z_F:{\mathrm{Bord}}_n\to {\mathrm{Cat}}_n(k)$ which sends a $k$-manifold $X$ to $F(X\times {ℝ}^{n-k})$, regarded as a bimodule between the analogous boundary restriction, and hence as a k-morphism in ${\mathrm{Cat}}_n(k)$.

From a ${\mathrm{Disk}}_n$-algebra $A$ we obtain the corresponding delooping $BA\in ({\mathrm{Cat}}_n(k{)}_{\mathrm{dualizable}}{)}^{O(n)}$ which is a $k$-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.

Examples

Dimension 1

A ${\mathrm{Disk}}_1$-algebra $A$ in ${\mathrm{Mod}}_k$ is equivalently a differential graded algebra.

The value of the corresponding $F_A\in H({\mathrm{Mfd}}_1,{\mathrm{Mod}}_k)$ on the circle is the Hochschild homology of $A$

$${\int }_{S^1}A\simeq {\int }_{{ℝ}^1}A{\otimes }_{{\int }_{S^0\times ℝ}A}{\int }_{{ℝ}^1}A\simeq {\mathrm{HH}}_{•}(A).$$\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 $n$-fold loop spaces

Given a topological space $Z$ we get a ${\mathrm{Disk}}_n$-algebra

$${\mathrm{Disk}}_n^{\coprod }\stackrel{{\mathrm{Maps}}_{\mathrm{compact}}(-,Z)}{\to }\mathrm{Top}\stackrel{C_{*}(-)}{\to }{\mathrm{Mod}}_k$$Disk_n^\coprod \stackrel{Maps_{compact}(-,Z)}{\to} Top \stackrel{C_\ast(-)}{\to} Mod_k

Where ${\mathrm{Maps}}_{\mathrm{compact}}({ℝ}^n,Z)\simeq {\Omega }^{n}Z$ is the n-fold loop space of $Z$.

Theorem (Salvatore and Lurie)

If $Z$ is $(n-1)$-n-connected object of an (infinity,1)-category

$${\int }_X{C}_{*}({\Omega }^{n}Z)\simeq C_{*}{\mathrm{Maps}}_{\mathrm{compact}}(X,Z).$$\int_X C_\ast(\Omega^n Z) \simeq C_\ast Maps_{compact}(X,Z) \,.

Related entries

• topological chiral homology, factorization algebra, blob homology

• local net of observables

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

References

The definition appears in section 3 of

• John Francis, The tangent complex and Hochschild cohomology of ${ℰ}_n$-rings (arXiv:1104.0181)

A detailed account is in

• John Francis, Factorization homology of topological manifolds (arXiv:1206.5522)

See also

• Hiro Lee Tanaka, Manifold calculus is dual to factorization homology, at Talbot 2012:_ Calculus of functors (pdf)

Application to higher Hochschild cohomology is discussed in

• Grégory Ginot, Thomas Tradler, Mahmoud Zeinalian, Higher Hochschild cohomology, Brane topology and centralizers of $E_n$-algebra maps, (arXiv:1205.7056)