assume that ${\mathcal{P}}$ is a pathwise connected poset equipped with a causal disjointness relation $\perp$. A *net of local algebras* indexed by ${\mathcal{P}}$ is a correspondence $${\mathscr{A}}_{\mathcal{P}}: {\mathcal{P}}\ni {\mathcal{O}}\longrightarrow {\mathcal{A}}({\mathcal{O}})\subseteq{\mathfrak{B}(\mathcal{H}_o)},$$ associating to any ${\mathcal{O}}$ a von Neumann algebras ${\mathcal{A}}({\mathcal{O}})$ defined on a fixed Hilbert space $\mathcal{H}_o$, and satisfying $$\begin{array}{lcll}
{\mathcal{O}}_1\leq {\mathcal{O}}_2 & \Rightarrow & {\mathcal{A}}({\mathcal{O}}_1)\subseteq {\mathcal{A}}({\mathcal{O}}_2), & 
\qquad isotony,\\
{\mathcal{O}}_1\perp {\mathcal{O}}_2   & \Rightarrow & {\mathcal{A}}({\mathcal{O}}_1)\subseteq {\mathcal{A}}({\mathcal{O}}_2)',& 
\qquad causality,
\end{array}$$ where the prime over the algebra stands for the commutant of the algebra. The algebra ${\mathcal{A}}({\mathcal{O}}^\perp)$ associated with the causal complement ${\mathcal{O}}^\perp$ of ${\mathcal{O}}\in{\mathcal{P}}$, is the $\mathrm{C}^*-$algebra generated by the algebras ${\mathcal{A}}({\mathcal{O}}_1)$ for any ${\mathcal{O}}_1\in{\mathcal{P}}$ with ${\mathcal{O}}_1\perp{\mathcal{O}}$. The net ${\mathscr{A}}_{\mathcal{P}}$ is said to be *irreducible* whenever, given $T\in{\mathfrak{B}(\mathcal{H}_o)}$ such that $T\in{\mathcal{A}}({\mathcal{O}})'$ for any ${\mathcal{O}}\in{\mathcal{P}}$, then $T=c\cdot \mathbbm{1}$.\
**The category of 1-cocycles -** We refer the reader to the Appendix for the definition of $\mathrm{C}^*-$category. Let ${\mathcal{P}}$ be a poset with a causal disjointness relation $\perp$, and let ${\mathscr{A}}_{\mathcal{P}}$ be an irreducible net of local algebras. A *1-cocycle* $z$ of ${\mathcal{P}}$ with values in ${\mathscr{A}}_{\mathcal{P}}$ is a field $z:{\Sigma}_1({\mathcal{P}})\ni b\longrightarrow z(b)\in {\mathfrak{B}(\mathcal{H}_o)}$ of unitary operators satisfying the 1-cocycle identity: $$z(\partial_0 c) \cdot z(\partial_2 c) =  z(\partial_1 c),  
\qquad  c\in{\Sigma}_2({\mathcal{P}}),$$ and the locality 