Homotopy Groups

Let X be a pointed fibrant simplicial set. The homotopy group \pi_n(X), as a set, is defined by \pi_n(X)=[S^n,X], i.e.  the set of the pointed homotopy classes of all pointed simplicial maps from S^n to X. \pi_n(X)=\pi_n(|X|) as sets.

An element x\in X_n is said to be spherical if d_i x=* for all 0\leq i\leq n.

Given a spherical element x\in X_n, then its representing map f_x:\Delta[n]\to X factors through the simplicial quotient set S^n=\Delta[n]/\partial\Delta[n]. Conversely, any simplicial map f:S^n\to X gives a spherical element f(\sigma_n)\in X_n, where \sigma_n is the nondegenerate element in S^n_n. This gives a one-to-one correspondence from the set of spherical elements in X_n to the set of simplicial maps S^n\to X.

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.

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