The affine characteristic singular simplex of an ordered simplex

Let $\sigma =[v_0,\dots ,v_n]$ be an $n$-simplex of an abstract simplicial complex $K$ together with an ordering of its vertices. Its geometric realization $\mathrm{\mid }\sigma \mathrm{\mid }$ is the simplex in $\mathrm{\mid }K\mathrm{\mid }$ spanned by those vertices (The geometric realization of an abstract simplicial complex).

The affine characteristic singular simplex of $\sigma$ is the unique affine map ${\chi }_{\sigma }:{\mathrm{\Delta }}^n\to \mathrm{\mid }K\mathrm{\mid }$ that sends the vertex $v_i$ of the standard simplex (The standard topological simplex and its affine face maps) to the vertex $v_i$ of $\mathrm{\mid }\sigma \mathrm{\mid }$ for each $i$. Its image is exactly $\mathrm{\mid }\sigma \mathrm{\mid }$.

Changing the chosen ordering by a permutation $\pi$ precomposes ${\chi }_{\sigma }$ with the affine self-map of ${\mathrm{\Delta }}^n$ that permutes its vertices by $\pi$. Different orderings therefore generally give different singular simplices, and the comparison map below removes this ambiguity by fixing a global vertex order on the simplicial complex.