Associativity and Path Inverse for Fundamental Groupoids

Continued from Path product and fundamental groupoids

(Associativity). Let $X$ be a fibrant simplicial set and let $\lambda_1$ , $\lambda_2$ and $\lambda_3$ be paths in $X$ such that $\lambda_1(1)=\lambda_2(0)$ and $\lambda_2(1)=\lambda_3(0)$ . Then $(\lambda_1*\lambda_2)*\lambda_3\simeq\lambda_1*(\lambda_2*\lambda_3)\ \text{rel}\ \partial\Delta[1]$

Let $y\in Y_0$ be a point. Denote $\epsilon_y:X\to Y$ as the constant simplicial map $\epsilon(x)=s_0^n(y)$ for $x\in X_n$ .

(Path Inverse). Let $X$ be a fibrant simplicial set and let $\lambda$ be a path in $X$ . Then there exists a path $\lambda^{-1}$ such that $\lambda*\lambda^{-1}\simeq\epsilon_{\lambda(0)}\ \text{rel}\ \partial\Delta[1]$ .