Homotopy Prefactorization Algebras

We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space $M$. This allows us to describe a notion of prefactorization algebra up to homotopy as well as morphisms up to homotopy between such objects. We make explicit these notions for several special $M$, such as certain finite topological spaces, or the real line.