On the equivalence of AQFTs and prefactorization algebras

This paper revisits the equivalence problem between algebraic quantum field theories and prefactorization algebras defined over globally hyperbolic Lorentzian manifolds. We develop a radically new approach whose main innovative features are 1.) a structural implementation of the additivity property used in earlier approaches and 2.) a reduction of the global equivalence problem to a family of simpler spacetime-wise problems. When applied to the case where the target category is a symmetric monoidal 1-category, this yields a generalization of the equivalence theorem from [Commun. Math. Phys. 377 , 971 (2019)]. In the case where the target is the symmetric monoidal ∞ ∞ -category of cochain complexes, we obtain a reduction of the global ∞ ∞ -categorical equivalence problem to simpler, but still challenging, spacetime-wise problems. The latter would be solved by showing that certain functors between 1-categories exhibit ∞ ∞ -localizations; however, the available detection criteria are inconclusive in our case.