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. Definition Prefactorization algebras A factorization algebra is a prefactorization algebra satisfying some properties, similar to sheafs being a presheaf with extra conditions. If is a topological space, a prefactorization algebra of vector spaces on is an assignment of vector spaces to open sets of , along with the following conditions on the assignment: For each inclusion , there's a linear map There is a linear map for each finite collection of open sets with each and the pairwise disjoint. The maps compose in the obvious way: for collections of opens , and an open satisfying and , the following diagram commutes. So 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 an open set, a collection of opens is a Weiss cover of if for any finite collection of points in , there is an open set such that . Then a factorization algebra of vector spaces on is a prefactorization algebra of vector spaces on so that for every open and every Weiss cover of , the sequence is exact. That is, 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 , the structure map is an isomorphism. Algebro-geometric formulation Let be a smooth complex curve. A factorization algebra on consists of A quasicoherent sheaf over for any finite set , with no non-zero local section supported at the union of all partial diagonals Functorial isomorphisms of quasicoherent sheaves over for surjections . (Factorization) Functorial isomorphisms of quasicoherent sheaves over . (Unit) Let and . A global section (the unit) with the property that for every local section (), the section of extends across the diagonal, and restricts to . Example Associative algebra Any associative algebra can be realized as a prefactorization algebra on . To each open interval , assign . An arbitrary open is a disjoint union of countably many open intervals, ,