Representing map of x

(Representing map of x). Let X be a simplicial set and let x\in X_n. There exists a unique simplicial map f_x:\Delta[n]\to X such that f_x(\sigma_n)=x.
Let (i_0,i_1,\dots,i_k)\in\Delta[n]. (i_0,i_1,\dots,i_k) can be written as iterated compositions of faces and degeneracies of \sigma_n, i.e. (i_0,i_1,\dots,i_k)=g\sigma_n where g is an iterated composition of faces (d_j) and degeneracies (s_j). Then
\begin{aligned}  f_x(i_0,i_1,\dots,i_k)&=f_x(g\sigma_n)\\  &=gf_x(\sigma_n)\\  &=gx  \end{aligned}
This defines a unique simplicial map f_x such that f_x(\sigma_n)=x.


About mathtuition88
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