# 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.

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## 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 $Mfd_n^{\coprod}$ 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

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

This inverse sends an $n$-disk algebra

$A : Disk_n \to Mod_k$

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

The factorization homology is then the derived coend

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

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

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

duality between algebra and geometry in physics:

## References

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