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 2011. 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 homology (Francis b).

Definition

Write $Mfd_n^{\coprod}$ 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 $Mod_k$ for the symmetric monoidal (∞,1)-category of $k$-chain complexes.

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

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

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

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

$$H(Mfd_n, Mod_k) \to Disk_n-Alg (Mod_k)$$

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

$$FactorizationHomology : Disk_n-Alg (Mod_k) \to H(Mfd_n, Mod_k) \,.$$

which sends an $n$-disk algebra $A : Disk_n \to Mod_k$ to the functor that sends a manifold $X$ to the derived coend

$$\int^X A = \mathbb{E}_X \otimes_{Disk_n} A$$

of $A$ with

$$\mathbb{E}_X : Disk_n \stackrel{Emb(-,X)}{\to} Top \stackrel{C_\bullet(-)}{\to} Mod_k \,.$$

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 $F \in H(Mfd_n, Mod_k)$ we get an extended TQFT with values in $k$-linear $(\infty,n)$-categories

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

From a $Disk_n$-algebra $A$ we obtain the corresponding delooping $\mathbf{B}A \in (Cat_n(k)_{dualizable})^{O(n)}$ 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 $Disk_1$-algebra $A$ in $Mod_k$ is equivalently a differential graded algebra.

The value of the corresponding $F_A \in H(Mfd_1, Mod_k)$ on the circle is the Hochschild homology of $A$

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

From $n$-fold loop spaces

Given a topological space $Z$ we get a $Disk_n$-algebra

$$Disk_n^\coprod \stackrel{Maps_{compact}(-,Z)}{\to} Top \stackrel{C_\ast(-)}{\to} Mod_k$$

Where $Maps_{compact}(\mathbb{R}^n, Z) \simeq \Omega^n Z$ 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

$$\int_X C_\ast(\Omega^n Z) \simeq C_\ast Maps_{compact}(X,Z) \,.$$

Related entries

• topological chiral homology, factorization algebra, blob homology

• local net of observables

• representation stability

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\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 }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

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

References

General

The definition appears in section 3 of

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

A detailed account is in

• David Ayala, John Francis, §4 in: Factorization homology of topological manifolds, Journal of Topology 8 4 (2015) 1045-1084 [arXiv:1206.5522, doi:10.1112/jtopol/jtv028]

A survey that also covers factorization algebras is

• Grégory Ginot, Notes on factorization algebras, factorization homology and applications, arXiv:1307.5213.

See also

• Jacob Lurie, Higher Algebra, section 5.3.

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

Generalization to orbifolds:

• Tim Weelinck, Equivariant factorization homology of global quotient orbifolds (arXiv:1810.12021, talk pdf)

Some applications are

• Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, arXiv:1405.6696.

• Quoc Ho, Densities and stability via factorization homology, arXiv:1802.07948.

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)

• Geoffroy Horel, Higher Hochschild homology of the Lubin-Tate ring spectrum, Algebr. Geom. Topol. 15 (2015) 3215-3252 [arXiv:1311.2805, doi:10.2140/agt.2015.15.3215]

Application to stratified spaces with tangential structures is discussed in

• David Ayala, John Francis, Hiro Lee Tanaka, Factorization homology of stratified spaces, (arXiv:1409.0848)

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

• David Ayala, John Francis, Poincaré/Koszul duality, arXiv:1409.2478.

Discussion in the context of extended TQFT appears in

• Claudia Scheimbauer, Factorization homology as a fully extended topological field theory (pdf)

• Videos from from the BIRS Workshop 15w5125 on Factorizable Structures in Topology and Algebraic Geometry.

For surfaces equipped with flat connections for a finite group:

• Corina Keller, Lukas Müller, Finite symmetries of quantum character stacks, Theory and Applications of Categories 39 3 (2023) 51-97 [arXiv:2107.12348, tac:39-03]

Survey:

• Corina Keller, Twisted Character Varieties and Quantization via Factorization Homology, talk at PIRSA (2022) [video: doi:10.48660/21100028]

• Lukas Müller, Deformation quantization and categorical factorization homology, talk at CQTS (Mar 2023) [web, video:YT]

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]