Our definition of a prefactorization algebra is closely related to that of a precosheaf or of a presheaf. Mathematicians have found it useful to refine the axioms of a presheaf to those of a sheaf: a sheaf is a presheaf whose value on a large open set is determined, in a precise way, by values on arbitrarily small subsets. In this chapter we describe a similar “descent” axiom for prefactorization algebras. We call a prefactorization algebra satisfying this axiom a factorization algebra.

After defining this axiom, our next task is to verify that the examples we have constructed so far, such as the observables of a free field theory, satisfy it. This we do in Sections 6.5 and 6.6.

Philosophically, our descent axiom for factorization algebras is important: a prefactorization algebra satisfying descent (i.e., a factorization algebra) is built from local data, in a way that a general prefactorization algebra need not be. However, for many practical purposes, such as the applications to field theory, this axiom is often not essential.

A reader with little taste for formal mathematics could thus skip this chapter and the next and still be able to follow the rest of this book.

Factorization Algebras

A factorization algebra is a prefactorization algebra that satisfies a local-toglobal axiom. This axiom is the analog of the gluing axiom for sheaves; it expresses how the values on big open sets are determined by the values on small open sets. We thus begin by reviewing the notion of a (co)sheaf, with more background and references available in Appendix A.5. Next, we introduce Weiss covers, which are the type of covers appropriate for factorization algebras. Finally, we give the definition of factorization algebras,

Sheaves and Cosheaves

In order to motivate our definition of factorization algebra, let us recall the sheaf axiom and then work out its dual, the cosheaf axiom.

Let M be a topological space and let Opens(M) denote the poset category of open subsets of a space M. A presheaf A on a topological space M with values in a category C is a functor from Opens(M)op to C.