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 applied in a more general setting by Costello and Gwilliam to formalize quantum field theory.

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:

${\displaystyle {\begin{array}{lcl}&\bigotimes _{i}\bigotimes _{j}{\mathcal {F}}(U_{i,j})&\rightarrow &\bigotimes _{i}{\mathcal {F}}(V_{i})&\\&\downarrow &\swarrow &\\&{\mathcal {F}}(W)&&&\\\end{array}}}$

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.

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 $${\displaystyle \bigoplus _{i,j}{\mathcal {F}}(U_{i}\cap U_{j})\rightarrow \bigoplus _{k}{\mathcal {F}}(U_{k})\rightarrow {\mathcal {F}}(U)\rightarrow 0}$$ is exact. That is, ${\displaystyle {\mathcal {F}}}$ is a factorization algebra if it is a cosheaf with respect to the Weiss topology.