\subseteq U$, for $i=1,2$. Since $K := \overline{U_1} \cup \overline{U_2}$ is compact, there exists a Cauchy surface $\Sigma$ of $M$ such that $K \subseteq I^-_M(\Sigma)$. We set $S := J^+_M(K) \cap J^-_M(\Sigma)$ and observe that this is a compact subset of $M$ by [@BGP Corollary A.5.4]. Using also [@BGP Lemma A.5.12], it follows that $U := I^+_M(K) \cap I^-_M(S)$ belongs to $\RC_M$. By construction, $U$ contains both $U_1$ and $U_2$.

We may restrict the orientation, time-orientation and metric on $M$ to the causally convex open subsets $U\in\RC_M$ and thereby define objects $U\in\Loc$. Every inclusion $U\subseteq V$ in $\RC_M$ then defines a $\Loc$-morphism $\iota_U^V : U\to V$. Hence, we can regard $\RC_M\subseteq \Loc$ as a subcategory, for every $M\in\Loc$, and restrict any prefactorization algebra $\FFF\in\PFA$ to a functor $\FFF\vert_{M} : \RC_M\to \CC$.

\[def:additivePFA\] A prefactorization algebra $\FFF\in\PFA$ is called [*additive*]{} if, for every $M\in\Loc$, the canonical morphism

\[eqn:additivePFA\] @C=3em[ (\_M : \_M) \^- & (M) ]{}

is an isomorphism in $\CC$. We denote by $\PFA^{\add}\subseteq \PFA$ the full subcategory of additive prefactorization algebras.

The additivity condition formalizes the idea that $\FFF(M)$ is “generated” by the images of the maps $\FFF(U)\to\FFF(M)$, for all relatively compact and causally convex open subsets $U\subseteq M$. 