all singular simplices of ${\mathcal{P}}$. ${\Sigma}_*({\mathcal{P}})$ is the *simplicial set* of ${\mathcal{P}}$. The inclusion maps $d^n_i$ between standard simplices are extended to maps $\partial^n_i :\Sigma_n({\mathcal{P}})\longrightarrow \Sigma_{n-1}({\mathcal{P}})$, called *boundaries*, between singular simplices by setting $\partial^n_i f  {\equiv}f \circ d^n_i$. One can easily check, by the definition of $d^n_i$, that the following relations $$\partial^{n-1}_i\circ \partial^{n}_j = 
\partial^{n-1}_j \circ \partial^{n}_{i+1},\qquad  i\geq j,$$ hold. From now on, we will omit the superscripts from the symbol $\partial^n_i$, and will denote: the composition $\partial_i\circ \partial_j$ by the symbol $\partial_{ij}$; 0-simplices by the letter $a$; 1-simplices by $b$ and 2-simplices by $c$. Notice that a 0-simplex $a$ is nothing but an element of ${\mathcal{P}}$; a 1-simplex $b$ is formed by two 0-simplices $\partial_0b$, $\partial_1b$ and an element $|b|$ of ${\mathcal{P}}$, called the *support* of $b$, such that $\partial_0b$, $\partial_1b\leq |b|$. Given $a_0,a_1\in{\Sigma}_0({\mathcal{P}})$, a *path from $a_0$ to $a_1$* is a finite ordered sequence $p=\{b_n,\ldots,b_1\}$ of 1-simplices satisfying the relations $$\partial_1b_1=a_0, \ \ \ \partial_0 b_{i} = \partial_1 b_{i+1} \  \mbox{ with } 
\ i\in\{1,\ldots,n-1\}, \ \ \  
                                          \partial_0b_n=a_1.$$ The *startingpoint* of $p$, written $\partial_1p$, is the 0-simplex $a_0$, while the *endpoint* of $p$, written $\partial_0p$, is the 0-simplex $a_1$. We will denote by $\mathrm{P}(a_0,a_1)$ the set of 