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,[1] and also studied in a more general setting by Costello to study quantum field theory.[2]
A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions.
If [math]\displaystyle{ M }[/math] is a topological space, a prefactorization algebra [math]\displaystyle{ \mathcal{F} }[/math] of vector spaces on [math]\displaystyle{ M }[/math] is an assignment of vector spaces [math]\displaystyle{ \mathcal{F}(U) }[/math] to open sets [math]\displaystyle{ U }[/math] of [math]\displaystyle{ M }[/math], along with the following conditions on the assignment:
[math]\displaystyle{ \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} }[/math]
So [math]\displaystyle{ \mathcal{F} }[/math] 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.
To define factorization algebras, it is necessary to define a Weiss cover. For [math]\displaystyle{ U }[/math] an open set, a collection of opens [math]\displaystyle{ \mathfrak{U} = \{U_i | i \in I\} }[/math] is a Weiss cover of [math]\displaystyle{ U }[/math] if for any finite collection of points [math]\displaystyle{ \{x_1, \cdots, x_k\} }[/math] in [math]\displaystyle{ U }[/math], there is an open set [math]\displaystyle{ U_i \in \mathfrak{U} }[/math] such that [math]\displaystyle{ \{x_1, \cdots, x_k\} \subset U_i }[/math].
Then a factorization algebra of vector spaces on [math]\displaystyle{ M }[/math] is a prefactorization algebra of vector spaces on [math]\displaystyle{ M }[/math] so that for every open [math]\displaystyle{ U }[/math] and every Weiss cover [math]\displaystyle{ \{U_i | i \in I\} }[/math] of [math]\displaystyle{ U }[/math], the sequence [math]\displaystyle{ \bigoplus_{i,j} \mathcal{F}(U_i \cap U_j) \rightarrow \bigoplus_k \mathcal{F}(U_k) \rightarrow \mathcal{F}(U) \rightarrow 0 }[/math] is exact. That is, [math]\displaystyle{ \mathcal{F} }[/math] 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 [math]\displaystyle{ U, V \subset M }[/math], the structure map [math]\displaystyle{ m^{U, V}_{U\sqcup V} : \mathcal{F}(U)\otimes \mathcal{F}(V) \rightarrow \mathcal{F}(U \sqcup V) }[/math] is an isomorphism.
While this formulation is related to the one given above, the relation is not immediate.
Let [math]\displaystyle{ X }[/math] be a smooth complex curve. A factorization algebra on [math]\displaystyle{ X }[/math] consists of
[math]\displaystyle{ j^*_{J/I}\mathcal{V}_{X, J} \rightarrow j^*_{J/I}(\boxtimes_{i \in I} \mathcal{V}_{X, p^{-1}(i)}) }[/math] over [math]\displaystyle{ U^{J/I} }[/math].
Any associative algebra [math]\displaystyle{ A }[/math] can be realized as a prefactorization algebra [math]\displaystyle{ A^{f} }[/math] on [math]\displaystyle{ \mathbb{R} }[/math]. To each open interval [math]\displaystyle{ (a,b) }[/math], assign [math]\displaystyle{ A^f((a,b)) = A }[/math]. An arbitrary open is a disjoint union of countably many open intervals, [math]\displaystyle{ U = \bigsqcup_i I_i }[/math], and then set [math]\displaystyle{ A^f(U) = \bigotimes_i A }[/math]. The structure maps simply come from the multiplication map on [math]\displaystyle{ A }[/math]. Some care is needed for infinite tensor products, but for finitely many open intervals the picture is straightforward.
Original source: https://en.wikipedia.org/wiki/Factorization algebra.
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