condition: $z(b)\in{\mathcal{A}}(|b|)$ for any 1-simplex $b$. An *intertwiner* $t\in(z,z_1)$ between a pair of 1-cocycles $z,z_1$ is a field $t:{\Sigma}_0({\mathcal{P}})\ni a\longrightarrow t_a\in{\mathfrak{B}(\mathcal{H}_o)}$ satisfying the relation $$t_{\partial_0b}\cdot z(b) =  z_1(b)\cdot t_{\partial_1b},
\qquad b\in{\Sigma}_1({\mathcal{P}}),$$ and the locality condition: $t_a\in{\mathcal{A}}(a)$ for any 0-simplex $a$. The *category of 1-cocycles* ${\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$ is the category whose objects are 1-cocycles and whose arrows are the corresponding set of intertwiners. The composition between $s\in(z,z_1)$ and $t\in(z_1,z_2)$ is the arrow $t\cdot s\in (z,z_2)$ defined as $$(t\cdot s)_a {\equiv}t_a\cdot s_a, \qquad a\in{\Sigma}_0({\mathcal{P}}).$$ Note that the arrow $1_z$ of $(z,z)$ defined as $(1_z)_a =\mathbbm{1}$, for any $a\in{\Sigma}_0({\mathcal{P}})$, is the identity of $(z,z)$. Now, the set $(z,z_1)$ has a structure of complex vector space defined as $$(\alpha\cdot t + {\beta}\cdot s)_a{\equiv}\alpha\cdot t_a + {\beta}\cdot s_a, \qquad 
 a\in{\Sigma}_0({\mathcal{P}}),$$ for any ${\alpha},{\beta}\in\mathbb{C}$ and $t,s\in(z,z_1)$. With these operations and the composition “$\cdot$”, the set $(z,z)$ is an algebra with identity $1_z$. The category ${\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$ has an adjoint $*$, defined on as the identity, $z^*=z$, on the objects, while the adjoint $t^*\in (z_1,z)$ of on arrows $t\in(z,z_1)$ is defined as $$(t^*)_a {\equiv}(t_a)^*, \qquad a\in{\Sigma}_0({\mathcal{P}}),$$ where $(t_a)^*$ stands for the adjoint in ${\mathfrak{B}(\mathcal{H}_o)}$ of the operator $t_a$. Now, let ${\Vert{ \ }\Vert}$ be the norm of ${\mathfrak{B}(\mathcal{H}_o)}$. Given 