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].

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