# Factorization algebra

In mathematics and mathematical physics, a factorization algebra is an algebraic structure first introduced by Beilinson and Drinfel'd in an algebro-geometric setting as a reformulation of chiral algebras, and also studied in a more general setting by Costello and Gwilliam to study quantum field theory.

## Definition
### Prefactorization algebras

A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.
If $M$ is a topological space, a prefactorization algebra ${\mathcal {F}}$ of vector spaces on $M$ is an assignment of vector spaces ${\mathcal {F}}(U)$ to open sets $U$ of $M$, along with the following conditions on the assignment:
* For each inclusion $U\subset V$, there's a linear map $m_{V}^{U}:{\mathcal {F}}(U)\rightarrow {\mathcal {F}}(V)$
* There is a linear map $m_{V}^{U_{1},\cdots ,U_{n}}:{\mathcal {F}}(U_{1})\otimes \cdots \otimes {\mathcal {F}}(U_{n})\rightarrow {\mathcal {F}}(V)$ for each finite collection of open sets with each $U_{i}\subset V$ and the $U_{i}$ pairwise disjoint.
* The maps compose in the obvious way: for collections of opens $U_{i,j}$, $V_{i}$ and an open $W$ satisfying $U_{i,1}\sqcup \cdots \sqcup U_{i,n_{i}}\subset V_{i}$ and $V_{1}\sqcup \cdots V_{n}\subset W$, the following diagram commutes.
${\begin{array}{lcl}&\bigotimes _{i}\bigotimes _{j}{\mathcal {F}}(U_{i,j})&\rightarrow &\bigotimes _{i}{\mathcal {F}}(V_{i})&\\&\downarrow &\swarrow &\\&{\mathcal {F}}(W)&&&\\\end{array}}$
So ${\mathcal {F}}$ resembles a precosheaf, except the vector spaces are tensored rather than (direct-)summed.
The category of vector spaces can be replaced with any symmetric monoidal category.

### Factorization algebras

To define factorization algebras, it is necessary to define a Weiss cover. For $U$ an open set, a collection of opens ${\mathfrak {U}}=\{U_{i}|i\in I\}$ is a Weiss cover of $U$ if for any finite collection of points $\{x_{1},\cdots ,x_{k}\}$ in $U$, there is an open set $U_{i}\in {\mathfrak {U}}$ such that $\{x_{1},\cdots ,x_{k}\}\subset U_{i}$.
Then a factorization algebra of vector spaces on $M$ is a prefactorization algebra of vector spaces on $M$ so that for every open $U$ and every Weiss cover $\{U_{i}|i\in I\}$ of $U$, the sequence $\bigoplus _{i,j}{\mathcal {F}}(U_{i}\cap U_{j})\rightarrow \bigoplus _{k}{\mathcal {F}}(U_{k})\rightarrow {\mathcal {F}}(U)\rightarrow 0$ is exact. That is, ${\mathcal {F}}$ is a factorization algebra if it is a cosheaf with respect to the Weiss topology.
A factorization algebra is multiplicative if, in addition, for each pair of disjoint opens $U,V\subset M$, the structure map $m_{U\sqcup V}^{U,V}:{\mathcal {F}}(U)\otimes {\mathcal {F}}(V)\rightarrow {\mathcal {F}}(U\sqcup V)$ is an isomorphism.

### Algebro-geometric formulation

While this formulation is related to the one given above, the relation is not immediate.
Let $X$ be a smooth complex curve. A factorization algebra on $X$ consists of
* A quasicoherent sheaf ${\mathcal {V}}_{X,I}$ over $X^{I}$ for any finite set $I$, with no non-zero local section supported at the union of all partial diagonals
* Functorial isomorphisms of quasicoherent sheaves $\Delta _{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow {\mathcal {V}}_{X,I}$ over $X^{I}$ for surjections $J\rightarrow I$.
* (Factorization) Functorial isomorphisms of quasicoherent sheaves
$j_{J/I}^{*}{\mathcal {V}}_{X,J}\rightarrow j_{J/I}^{*}(\boxtimes _{i\in I}{\mathcal {V}}_{X,p^{-1}(i)})$ over $U^{J/I}$.
* (Unit) Let ${\mathcal {V}}={\mathcal {V}}_{X,\{1\}}$ and ${\mathcal {V}}_{2}={\mathcal {V}}_{X,\{1,2\}}$. A global section (the unit) $1\in {\mathcal {V}}(X)$ with the property that for every local section $f\in {\mathcal {V}}(U)$ ($U\subset X$), the section $1\boxtimes f$ of ${\mathcal {V}}_{2}|_{U^{2}\Delta }$ extends across the diagonal, and restricts to $f\in {\mathcal {V}}\cong {\mathcal {V}}_{2}|_{\Delta }$.

## Example
### Associative algebra

Any associative algebra $A$ can be realized as a prefactorization algebra $A^{f}$ on $\mathbb {R}$. To each open interval $(a,b)$, assign $A^{f}((a,b))=A$. An arbitrary open is a disjoint union of countably many open intervals, $U=\bigsqcup _{i}I_{i}$, and then set $A^{f}(U)=\bigotimes _{i}A$. The structure maps simply come from the multiplication map on $A$. Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.