$t\in(z,z_1)$, we have that ${\Vert{t_a}\Vert}={\Vert{t_{a_1}}\Vert}$ for any pair $a,a_1$ of 0-simplices because ${\mathcal{P}}$ is pathwise connected. Therefore, by defining $${\Vert{t}\Vert} {\equiv}{\Vert{t_a}\Vert} \qquad a\in{\Sigma}_0({\mathcal{P}})$$ it turns out $(z,z_1)$ is a complex Banach space for any $z,z_1\in{\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$, while $(z,z)$ is a $\mathrm{C}^*-$algebra for any $z\in{\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$. This entails that ${\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$ is a $\mathrm{C}^*-$category. Two 1-cocycles $z,z_1$ are *equivalent* (or *cohomologous*) if there exists a unitary arrow $t\in(z,z_1)$. A 1-cocycle $z$ is *trivial* if it is equivalent to the *identity* cocycle ${\iota}$ defined as ${\iota}(b)=\mathbbm{1}$ for any 1-simplex $b$. Note that, since ${\mathscr{A}}_{{\mathcal{P}}}$ is irreducible, ${\iota}$ is irreducible: $({\iota},{\iota})= \mathbb{C}\mathbbm{1}$.\
**Equivalence in $\boldsymbol{{\mathfrak{B}(\mathcal{H}_o)}}$ and path-independence -** A weaker form of equivalence between 1-cocycles is the following: $z,z_1$ are said to be *equivalent in* ${\mathfrak{B}(\mathcal{H}_o)}$ if there exists a field $V:{\Sigma}_0({\mathcal{P}})\ni a \longrightarrow V_a\in{\mathfrak{B}(\mathcal{H}_o)}$ of unitary operators such that $$V_{\partial_0b}\cdot z(b) =  z_1(b)\cdot V_{\partial_1b}, \qquad 
b\in{\Sigma}_1({\mathcal{P}}).$$ Note that the field $V$ is not an arrow of $(z,z_1)$ because it is not required that $V$ satisfies the locality condition. A 1-cocycle is *trivial in* ${\mathfrak{B}(\mathcal{H}_o)}$ if it is equivalent in ${\mathfrak{B}(\mathcal{H}_o)}$ to the trivial 1-cocycle ${\iota}$. We denote by ${\mathcal{Z}^1_t({\mathscr{A}}_{\mathcal{P}})}$ the set of the 1-cocycles trivial in ${\mathfrak{B}(\mathcal{H}_o)}$ and with the same symbol we denote the 