This paper develops a novel approach to functorial quantum field theories
(FQFTs) in the context of Lorentzian geometry. The key challenge is that
globally hyperbolic Lorentzian bordisms between two Cauchy surfaces cannot
change the topology of the Cauchy surface. This is addressed and solved by
introducing a more flexible concept of bordisms which provide morphisms from
tuples of causally disjoint partial Cauchy surfaces to a later-in-time full
Cauchy surface. They assemble into a globally hyperbolic Lorentzian bordism
pseudo-operad, generalizing the geometric bordism pseudo-categories of Stolz
and Teichner. The associated FQFTs are defined as pseudo-multifunctors into a
symmetric monoidal category of unital associative algebras. The main result of
this paper is an equivalence theorem between such globally hyperbolic
Lorentzian FQFTs and algebraic quantum field theories (AQFTs), both subject to
the time-slice axiom and a mild descent condition called additivity.

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