Functorial Field Theories and Factorization Algebras

ing the data and the properties of our motivating example, we now define prefactorization algebras and strict factorization algebras. In the example above, we associate to each open subset U a vector space. This is the main example we are interested in, but more generally, we will define (pre) factorization algebras with values in a monoidal category (C,⊗). Definition 4.18. Let M be a topological space. A prefactorization algebra F on M with values in a monoidal category (C,⊗) assigns • to each open subset U ⊂M a vector space F(U), and • to a collection U1, . . . , Uk of disjoint open subsets of an open subset V ⊂ M a linear map m1k U : F(U1)⊗ · · · ⊗ F(Uk) −→ F(V ). These maps are required to be compatible with composition in the sense that if Ui1, . . . , Uini are disjoint open subsets of Vi, and V1, . . . , Vk are disjoint open subsets of W , then the diagram k ⊗