Shapes of a simplicial complex

Question

In Bridson and Haefliger's book, page 98, there is a definition of shape. Here is a link to the book. The definiton is not very clear to me. It says set of isometry classes: Is it the isometry $h_{\lambda,\lambda'}$ ? Can anyone please give me a better explanation or reference?

Answer 1

A $M_{\kappa}$-simplicial complex $K$ can be viewed as a collection of geodesic simplices $\{ S_{\lambda} \mid \lambda \in \Lambda \}$ glued together by isometries; let $\mathcal{S}$ denote the set of all faces of the $S_{\lambda}$'s. The relation of isometry $\simeq$ on $\mathcal{S}$, that is $S_{\lambda_1} \simeq S_{\lambda_2}$ if they are isometric, is an equivalence relation. The shapes of $K$ are the equivalence classes of $\mathcal{S}$ modulo $\simeq$.

For example, let $K= \mathbb{R}^2$ viewed as a square complex (structure induced by $\mathbb{Z}^2$). Then the shapes of $K$ are: a dot $\{0 \}$, a segment $[0,1]$, and a square $[0,1]^2$.