# Haag-Kastler stacks

April 22, 2024

This paper provides an alternative implementation of the principle of general
local covariance for algebraic quantum field theories (AQFTs) which is more
flexible and powerful than the original one by Brunetti, Fredenhagen and Verch.
This is realized by considering the $2$-functor $\mathsf{HK} :
\mathbf{Loc}^\mathrm{op} \to \mathbf{CAT}$ which assigns to each Lorentzian
manifold $M$ the category $\mathsf{HK}(M)$ of Haag-Kastler-style AQFTs over $M$
and to each embedding $f:M\to N$ a pullback functor $f^\ast = \mathsf{HK}(f) :
\mathsf{HK}(N) \to \mathsf{HK}(M)$ restricting theories from $N$ to $M$.
Locally covariant AQFTs are recovered as the points of the $2$-functor
$\mathsf{HK}$. 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 $\mathbf{Loc}$ 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 $M$, are reintroduced. 3.) It provides a successful and 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 $M$ from its
local data in an open cover $\{U_i \subseteq M\}$.

Keywords:

algebraic quantum field theory, locally covariant quantum field theory, Lorentzian geometry, stacks, descent conditions, locally presentable categories