full $\mathrm{C}^*-$subcategory of ${\mathcal{Z}^1({\mathscr{A}}_{\mathcal{P}})}$ whose objects are the 1-cocycles trivial in ${\mathfrak{B}(\mathcal{H}_o)}$. Triviality in ${\mathfrak{B}(\mathcal{H}_o)}$ is related to the notion of path-independence. The evaluation of a 1-cocycle $z$ on a path $p=\{b_n,\ldots,b_1\}$ is defined as $$z(p){\equiv}z(b_n)\cdots z(b_2)\cdot z(b_1).$$ $z$ is said to be *path-independent* on a subset $P\subseteq{\mathcal{P}}$ whenever $$\label{Aa:1}
 z(p)=z(q)  \mbox{ for any } p,q\in\mathrm{P}(a_0,a_1) \mbox{ such that }
 |p|,|q|\subseteq P.$$ As ${\mathcal{P}}$ is pathwise connected, a 1-cocycle is trivial in ${\mathfrak{B}(\mathcal{H}_o)}$ if, and only if, it is path-independent on all ${\mathcal{P}}$ [@GLRV]. For later purposes, we recall the following result: assume that $z$ is a 1-cocycle trivial in ${\mathfrak{B}(\mathcal{H}_o)}$, if the causal complement ${\mathcal{O}}^\perp$ of ${\mathcal{O}}$ is pathwise connected then $$\label{Aa:2}
 z(p)\cdot A\cdot z(p)^* = A \qquad A\in{\mathcal{A}}({\mathcal{O}})$$ for any path $p$ with $\partial_1p, \partial_0p\perp {\mathcal{O}}$ [@GLRV Lemma 3A.5].

The first homotopy group of a poset {#Ab}
-----------------------------------

The logical steps necessary to define the first homotopy group of posets are the same as in the case of topological spaces. We first recall the definition of a homotopy of paths; secondly, we introduce the reverse of a path, the composition of paths and prove that they behave well under the homotopy equivalence relation; finally we define the first 