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.
symmetric monoidal (∞,1)-category of spectra
AQFT and operator algebra
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).
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
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
This turns out to be an equivalence of (∞,1)-categories. The inverse is defined to be factorization homology
This inverse sends an $n$-disk algebra
to the functor that sends a manifold $X$ to the
The factorization homology is then the derived coend
of $A$ with
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.
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.
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 $A$
Given a topological space $Z$ we get a $Disk_n$-algebra
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
duality between algebra and geometry in physics:
The definition appears in section 3 of
A detailed account is in
A survey that also covers factorization algebras is
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)
Some applications are
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, pdf.
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.