\sim \ b(a_0),\end{aligned}$$ completing the proof.

Connection between homotopy and net-cohomology {#Ac}
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Let us consider a pathwise-connected poset ${\mathcal{P}}$, equipped with a causal disjointness relation $\perp$, and let ${\mathscr{A}}_{\mathcal{P}}$ be an irreducible net of local algebras. In this section we show the relation between $\pi_1({\mathcal{P}})$ and the set ${\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$.\
To begin with, we prove the invariance of 1-cocycles for homotopic paths.

\[Ac:1\] Let $z\in{\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$. For any pair $p,q$ of paths with the same endpoints, if $p\sim q$, then $z(p)=z(q)$.

It is enough to check the invariance of $z$ for elementary deformations. For instance let $q=\{b_n, \ldots, b_{j+1},\partial_0c,\partial_2c,b_{j-1},\ldots, b_2, b_1\}$, with $c\in {\mathsf{A}}(b_j)$, that is an elementary ampliation of $p$. By definition of ${\mathsf{A}}(b_j)$ and the 1-cocycle identity we have $$\begin{gathered}
z(q)  = z(b_n)\cdots  z(b_{j+1}) \cdot z(\partial_0c)
  \cdot z(\partial_2c)\cdot z(b_{j-1})\cdots z(b_1) \\
  = z(b_n)\cdots  z(b_{j+1}) \cdot z(\partial_1c)\cdot z(b_{j-1})
    \cdots z(b_1) = z(p).\end{gathered}$$ The invariance for elementary contractions follows in a similar way.

\[Ac:2\] Let $z\in{\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$. Then:\
(a) $z(b(a))=\mathbbm{1}$ for any 0-simplex $a$;\
(b) $z({\overline{p}})=z(p)^*$ for any path $p$.

\(a) Let $c(a_0)$ be the 2-simplex degenerate to $a_0$, that is $$\partial_0c(a_0)=\partial_2c(a_0) =  \partial_1c(a_0) = b(a_0), \ \ 
          |c(a_0)|=a_0$$ where $b(a_0)$ is the 1-simplex degenerate to $a_0$. By the 1-cocycle identity we 