Haag–Kastler stacks

This paper provides an alternative implementation of the principle of general local covariance for algebraic quantum field theories (AQFTs) which is more flexible than the original one by Brunetti, Fredenhagen and Verch. This is realized by considering the [Formula: see text]-functor [Formula: see text] which assigns to each Lorentzian manifold [Formula: see text] the category [Formula: see text] of Haag–Kastler-style AQFTs over [Formula: see text] and to each embedding [Formula: see text] a pullback functor [Formula: see text] restricting theories from [Formula: see text] to [Formula: see text]. Locally covariant AQFTs are recovered as the points of the [Formula: see text]-functor [Formula: see text]. The main advantages of this new perspective are: (1) It leads to technical simplifications, in particular with regard to the time-slice axiom, since global problems on [Formula: see text] become families of simpler local problems on individual Lorentzian manifolds. (2) Some aspects of the Haag–Kastler framework which previously got lost in locally covariant AQFT, such as a relative compactness condition on the open subsets in a Lorentzian manifold [Formula: see text], are reintroduced. (3) It provides a radically new perspective on descent conditions in AQFT, i.e. local-to-global conditions which allow one to recover a global AQFT on a Lorentzian manifold [Formula: see text] from its local data in an open cover [Formula: see text].