Factorization algebra

Definition

Prefactorization algebras

A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.

If ${\displaystyle M}$ is a topological space, a prefactorization algebra ${\displaystyle \mathcal{F}}$ of vector spaces on ${\displaystyle M}$ is an assignment of vector spaces ${\displaystyle \mathcal{F}(U)}$ to open sets ${\displaystyle U}$ of ${\displaystyle M}$, along with the following conditions on the assignment:

• For each inclusion ${\displaystyle U\subset V}$, there's a linear map ${\displaystyle m_V^U:\mathcal{F}(U)\to \mathcal{F}(V)}$
• There is a linear map ${\displaystyle m_V^{U_1,\cdots ,U_n}:\mathcal{F}(U_1)\otimes \cdots \otimes \mathcal{F}(U_n)\to \mathcal{F}(V)}$ for each finite collection of open sets with each ${\displaystyle U_i\subset V}$ and the ${\displaystyle U_i}$ pairwise disjoint.
• The maps compose in the obvious way: for collections of opens ${\displaystyle U_{i,j}}$, ${\displaystyle V_i}$ and an open ${\displaystyle W}$ satisfying ${\displaystyle U_{i,1}\bigsqcup \cdots \bigsqcup U_{i,n_i}\subset V_i}$ and ${\displaystyle V_1\bigsqcup \cdots V_n\subset W}$, the following diagram commutes.

${\displaystyle \begin{array}{lclll}& \underset{i}{⨂}\underset{j}{⨂}\mathcal{F}(U_{i,j})& \to & \underset{i}{⨂}\mathcal{F}(V_i)& \\ & \downarrow & \swarrow & \\ & \mathcal{F}(W)& & & \end{array}}$

So ${\displaystyle \mathcal{F}}$ 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 ${\displaystyle U}$ an open set, a collection of opens ${\displaystyle \mathfrak{U}=\{U_i|i\in I\}}$ is a Weiss cover of ${\displaystyle U}$ if for any finite collection of points ${\displaystyle \{x_1,\cdots ,x_k\}}$ in ${\displaystyle U}$, there is an open set ${\displaystyle U_i\in \mathfrak{U}}$ such that ${\displaystyle \{x_1,\cdots ,x_k\}\subset U_i}$.

Then a factorization algebra of vector spaces on ${\displaystyle M}$ is a prefactorization algebra of vector spaces on ${\displaystyle M}$ so that for every open ${\displaystyle U}$ and every Weiss cover ${\displaystyle \{U_i|i\in I\}}$ of ${\displaystyle U}$, the sequence $${\displaystyle \underset{i,j}{⨁}\mathcal{F}(U_i\cap U_j)\to \underset{k}{⨁}\mathcal{F}(U_k)\to \mathcal{F}(U)\to 0}$$ is exact. That is, ${\displaystyle \mathcal{F}}$ 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 ${\displaystyle U,V\subset M}$, the structure map $${\displaystyle m_{U\bigsqcup V}^{U,V}:\mathcal{F}(U)\otimes \mathcal{F}(V)\to \mathcal{F}(U\bigsqcup V)}$$ is an isomorphism.

Algebro-geometric formulation

While this formulation is related to the one given above, the relation is not immediate.

Let ${\displaystyle X}$ be a smooth complex curve. A factorization algebra on ${\displaystyle X}$ consists of

• A quasicoherent sheaf ${\displaystyle {\mathcal{V}}_{X,I}}$ over ${\displaystyle X^I}$ for any finite set ${\displaystyle I}$, with no non-zero local section supported at the union of all partial diagonals
• Functorial isomorphisms of quasicoherent sheaves ${\displaystyle {\mathrm{\Delta }}_{J/I}^{\ast }{\mathcal{V}}_{X,J}\to {\mathcal{V}}_{X,I}}$ over ${\displaystyle X^I}$ for surjections ${\displaystyle J\to I}$.
• (Factorization) Functorial isomorphisms of quasicoherent sheaves

$${\displaystyle j_{J/I}^{\ast }{\mathcal{V}}_{X,J}\to j_{J/I}^{\ast }({⊠}_{i\in I}{\mathcal{V}}_{X,p^{-1}(i)})}$$ over ${\displaystyle U^{J/I}}$.

• (Unit) Let ${\displaystyle \mathcal{V}={\mathcal{V}}_{X,\{1\}}}$ and ${\displaystyle {\mathcal{V}}_2={\mathcal{V}}_{X,\{1,2\}}}$. A global section (the unit) ${\displaystyle 1\in \mathcal{V}(X)}$ with the property that for every local section ${\displaystyle f\in \mathcal{V}(U)}$ (${\displaystyle U\subset X}$), the section ${\displaystyle 1⊠f}$ of ${\displaystyle {\mathcal{V}}_2{|}_{U^2\mathrm{\Delta }}}$ extends across the diagonal, and restricts to ${\displaystyle f\in \mathcal{V}\cong {\mathcal{V}}_2{|}_{\mathrm{\Delta }}}$.