Canonical barycentric realization map

Write $e_v$ for the coordinate vertex of $\mathrm{\mid }K\mathrm{\mid }$. For each nonempty face $\sigma$ put $b_{\sigma }=\frac{1}{\mathrm{\#}\sigma }{\sum }_{v\in \sigma }{e}_v$. The canonical barycentric realization map is $b_K:\mathrm{\mid }\mathrm{sd}K\mathrm{\mid }⟶\mathrm{\mid }K\mathrm{\mid },b_K\left({\sum }_{i=0}^q{t}_i{e}_{{\sigma }_i}\right)={\sum }_{i=0}^q{t}_i{b}_{{\sigma }_i}\mathrm{.}$ Here $t_i\ge 0$, ${\sum }_i{t}_i=1$ and ${\sigma }_0⊊\cdots ⊊{\sigma }_q$, as in Barycentric subdivision of an abstract simplicial complex. Each coordinate is nonnegative, the total is $1$, and the support lies in ${\sigma }_q$, so the formula belongs to the realization in The geometric realization of an abstract simplicial complex. Zero coefficients can be deleted without changing the sum, so formulas on intersecting simplices agree. Each restriction is affine and continuous into the maximal original simplex, hence into $\mathrm{\mid }K\mathrm{\mid }$; the weak topology of the source makes $b_K$ continuous. For a vertex-free complex it is the unique empty map.

2.5.7–2.5.10, pp.49–52.