1. Example. The following image depicts the geometric realization and the associated height function of an \(\{a_1, a_2, a_3\}\)-skeleton for an \(\overline{{\mathbb{R}}}\)-space.

     

Now let \(S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}\) for some non-negative integer \(n\) and let \(a_0 = -\infty\) and \(a_{n+1} = \infty\).

  1. Definition. Let \(X\) and \(X'\) be two \(S\)-skeletons for an \(\overline{{\mathbb{R}}}\)-space and suppose we have the following data:

    • For \(i = 1, \dots, n\) a continuous map \(\varphi^v_i \colon V_i \rightarrow V'_i\).

    • For \(i = 0, \dots, n\) a continuous map \(\varphi^e_i \colon E_i \rightarrow E'_i\).

    This data describes a homomorphism of \(S\)-skeletons for an \(\overline{{\mathbb{R}}}\)-space \(\varphi \colon X \rightarrow X'\), if the two diagrams \[ \xymatrix{ V_i \ar[d]_{\varphi^v_i} & E_i \ar[d]^{\varphi^e_i} \ar[l]_{l_i} \\ V'_i & E'_i \ar[l]^{l'_i} } \] and \[ \xymatrix{ E_{i-1} \ar[r]^{r_{i-1}} \ar[d]_{\varphi^e_{i-1}} & V_i \ar[d]^{\varphi^v_i} \\ E'_{i-1} \ar[r]_{r'_{i-1}} & V'_i } \] commute for \(i = 1, \dots, n\).

    The composition of two homomorphisms of \(S\)-skeletons is defined by composing the individual maps \(\varphi^v_i\) and \(\varphi^e_i\).

In the first definition we described the geometric realization of an \(S\)-skeleton for an \(\overline{{\mathbb{R}}}\)-space. This picture is only complete, if we also describe a geometric pendant to any homomorphism between \(S\)-skeletons.

Altogether this defines a faithful functor \(|\_|\) from the category of \(S\)-skeletons for \(\overline{{\mathbb{R}}}\)-spaces to the category of \(\overline{{\mathbb{R}}}\)-spaces.