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).

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One Response to Path product and fundamental groupoids

  1. Pingback: Associativity and Path Inverse for Fundamental Groupoids | Singapore Maths Tuition

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