assume that ${\mathcal{P}}$ is pathwise connected. Given a 0-simplex $a_1$, let $q$ be a path from $a_0$ to $a_1$. Then the map $$\pi_1({\mathcal{P}},a_0)\ni [p]\longrightarrow  [q*p*{\overline{q}}]\in \pi_1({\mathcal{P}},a_1)$$ is a group isomorphism. On the grounds of these facts, we give the following

\[Ab:12\] We call $\pi_1({\mathcal{P}},a_0)$ the **first homotopy group** of ${\mathcal{P}}$ with base $a_0\in{\Sigma}_0({\mathcal{P}})$. If ${\mathcal{P}}$ is pathwise connected, we denote this group by $\pi_1({\mathcal{P}})$ and call it the **fundamental group** of ${\mathcal{P}}$. If $\pi_1({\mathcal{P}})=1$ we will say that ${\mathcal{P}}$ is **simply connected**.

We have the following result

\[Ab:13\] If ${\mathcal{P}}$ is directed, then ${\mathcal{P}}$ is pathwise and simply connected.

Clearly ${\mathcal{P}}$ is pathwise connected. Let $p= \{b_n,\ldots,b_1\} \in\mathrm{P}(a_0)$. As ${\mathcal{P}}$ is directed, we can find $c_i\in{\mathsf{A}}(b_i)$ for $i =2,\ldots, n-1$ with $$\partial_2c_2 = {\overline{b_1}}, \ \ 
 {\overline{\partial_0c_{i-1}}} = \partial_2c_{i} \ \mbox{ for } \ 
  i = 3,\ldots, n-1, \ \ \partial_0c_{n-1} =  {\overline{b_n}}.$$ One can easily deduce from these relations that $\partial_{02}c_i= a_0$ for any $i=2,\ldots, n-1$. By Lemmas \[Ab:5\], \[Ab:8\] and \[Ab:10\], we have $$\begin{aligned}
p & \sim b_n* \partial_0c_{n-1} * \partial_2c_{n-1}* 
            \partial_0c_{n-2} 
              *\cdots *  \partial_2c_{3}* 
\partial_0c_2 * \partial_2c_2* b_1 \\
 &  =  b_n * {\overline{b_n}}* \partial_2c_{n-1}* {\overline{\partial_2c_{n-1}}}*
  \cdots * \partial_2c_3*{\overline{\partial_2c_3}}*{\overline{b_1}}* b_1\\
  &  \sim b(a_0) * \cdots *b(a_0) \ 