In Section \[subsec:KleinGordon\], we apply our general results to the simple example given by the free Klein-Gordon field $\AAA_{\mathrm{KG}}\in\AQFT^{\addc}$. We observe as in [@GwilliamRejzner] that the corresponding time-orderable prefactorization algebra $\FFF_{\mathrm{KG}}\in\toPFA^{\addc}$ describes the time-ordered products from perturbative algebraic quantum field theory, cf. [@FredenhagenRejzner; @Rejzner].

\[sec:prel\]Preliminaries
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\[subsec:Lorentzian\]Lorentzian geometry
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In order to fix our notations, we shall briefly recall some basic definitions and properties of Lorentzian manifolds. We refer to [@BGP] for a concise introduction.

A [*Lorentzian manifold*]{} is a manifold $M$ together with a metric $g$ of signature $(-+\cdots+)$. A non-zero tangent vector $0\neq v\in T_x M$ at a point $x\in M$ is called [*time-like*]{} if $g(v,v)<0$, [*light-like*]{} if $g(v,v) =0$ and [*space-like*]{} if $g(v,v)>0$. It is called [*causal*]{} if it is either time-like or light-like, i.e. $g(v,v)\leq 0$. A curve $\gamma : I\to M$, where $I\subseteq \bbR$ is an open interval, is called [*time-like*]{}/[*light-like*]{}/[*space-like*]{}/[*causal*]{} if all its tangent vectors $\dot{\gamma}$ are time-like/light-like/space-like/causal. A Lorentzian manifold is called [*time-orientable*]{} if there exists a vector field $\mathfrak{t}\in\Gamma^\infty(TM)$ that is everywhere time-like. Such $\mathfrak{t}$ determines a [*time-orientation*]{}.

In what follows we always consider time-oriented Lorentzian manifolds, denoted collectively by symbols like $M$, suppressing the metric $g$ and time-orientation $\mathfrak{t}$ from our 