Constructible Spaces over the Reals

  • Definition. Let \(S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}\) for some non-negative integer \(n\) and let \(a_0 = -\infty\) and \(a_{n+1} = \infty\).
    Then an \(S\)-skeleton \(X\) for an \(\overline{{\mathbb{R}}}\)-space is given by the following data:

    • For \(i = 1, \dots, n\) a locally path-connected compact topological space \(V_i\).

    • For \(i = 0, \dots, n\) a locally path-connected compact topological space \(E_i\).

    • For \(i = 1, \dots, n\) two continuous maps \(l_i \colon E_i \rightarrow V_i\) and \(r_{i-1} \colon E_{i-1} \rightarrow V_i\).

    For an \(S\)-skeleton \(X\) as above we define the geometric realization \(|X|\) to be \[\left( \coprod_{i=1}^n V_i \times \{a_i\} \right) \coprod \coprod_{i=0}^{n} E_i \times [a_i, a_{i+1}] / \sim,\] where \(\sim\) is the equivalence relation generated by \((l_i(x), a_i) \sim (x, a_i)\) for \(i = 1, \dots, n\) and \((r_i(x), a_{i+1}) \sim (x, a_{i+1})\) for \(i = 0, \dots, n-1\). Moreover we define \(f_X \colon |X| \rightarrow \overline{{\mathbb{R}}}\) to be the map induced by the projection onto the second factor.

    An \(\overline{{\mathbb{R}}}\)-space given by a continuous map \(g \colon Y \rightarrow \overline{{\mathbb{R}}}\) is constructible if there is a finite subset \(S \subset {\mathbb{R}}\) and an \(S\)-skeleton \(X\), such that \(f_X \cong g\) as \(\overline{{\mathbb{R}}}\)-spaces.

    The \(S\)-skeleton \(X\) is an \(S\)-skeleton for a bounded \({\mathbb{R}}\)-space if \(E_0 = \emptyset = E_n\).

    A bounded \({\mathbb{R}}\)-space \(g \colon Y \rightarrow {\mathbb{R}}\) is constructible if there is a finite subset \(S \subset {\mathbb{R}}\) and an \(S\)-skeleton \(X\) with \(E_0 = \emptyset = E_n\), such that \(f_X \cong g\) as \(\overline{{\mathbb{R}}}\)-spaces.

  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.

  • Definition. For a homomorphism \(\varphi \colon X \rightarrow X'\) of \(S\)-skeletons for \(\overline{{\mathbb{R}}}\)-spaces, we define the geometric realization \(|\varphi|\) of \(\varphi\) to be the map induced on the quotients defined by the maps:

    • \(V_i \times \{a_i\} \rightarrow V'_i \times \{a_i\}, (p, a_i) \mapsto (\varphi^v_i (p), a_i)\) for \(i = 1, \dots, n\).

    • \(E_i \times [a_i, a_{i+1}] \rightarrow E'_i \times [a_i, a_{i+1}], (p, t) \mapsto (\varphi^e_i(p), t)\) for \(i = 0, \dots, n\).

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.