## Path product and fundamental groupoids

Let $\sigma_1=(0,1)\in\Delta[1]_1$. A path is a simplicial map $\lambda:\Delta[1]\to X$. Since $\text{Hom}(\Delta[1],X)\cong X_1$, the paths are in one-to-one correspondence to the elements in $X_1$ via the function $\lambda\mapsto x_\lambda=\lambda(\sigma_1)$. The initial point of $\lambda$ is $\lambda(0)=\lambda(d_1\sigma_1)=d_1x_\lambda\in X_0$, and the end point of $\lambda$ is $\lambda(1)=\lambda(d_0\sigma_1)=d_0x_\lambda\in X_0$.

Let $\lambda$ and $\mu$ be two paths such that $\lambda(1)=\mu(0)$. Then $d_0x_\lambda=d_1x_\mu$ and thus the elements $x_0=x_\mu$ and $x_2=x_\lambda$ have matching faces (with respect to 1).