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 to study quantum field theory.

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: $$ \begin{array}{lcl} & \bigotimes_i \bigotimes_j \mathcal{F}(U_{i,j}) & \rightarrow & \bigotimes_i \mathcal{F}(V_i) & \\ & \downarrow & \swarrow & \\ & \mathcal{F}(W) & &  & \\ \end{array} $$
 * 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.

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

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 $$ m^{U, V}_{U\sqcup V} : \mathcal{F}(U)\otimes \mathcal{F}(V) \rightarrow \mathcal{F}(U \sqcup V)$$ 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 over $$U^{J/I}$$.
 * 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
 * (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 $$f \in \mathcal{V} \cong \mathcal{V}_2|_\Delta$$.

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.