Graphs over the Reals

  • Definition. Let \(S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}\) for some non-negative integer \(n\). Then an \(S\)-graph \(G\) is given by the following data:

    • For \(i = 1, \dots, n\) a finite set \(V_i\).

    • For \(i = 0, \dots, n\) a finite set \(E_i\).

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

Thinking of such sets \(V_i\) as vertices and the sets \(E_i\) as edges, \(G\) can (almost) be seen as a multigraph.

  • Definition. Let \(S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}\) for some non-negative integer \(n\) and let \(X\) be an \(S\)-skeleton for an \(\overline{{\mathbb{R}}}\)-space, then we obtain an \(S\)-graph \(\mathcal{C} X\) by applying the path-connected components functor \(\pi_0\) to all spaces and maps defining \(X\). Moreover we may apply \(\pi_0\) to all the individual maps \(\varphi^v_i\) and \(\varphi^e_i\) describing a homomorphism \(\varphi\) of \(S\)-skeletons, to obtain a functor \(\mathcal{C}\) from the category of \(S\)-skeletons to the category of \(S\)-graphs.

  1. Example. If \(X\) is the \(\{a_1, a_2, a_3\}\)-skeleton depicted in example 27, then the following image shows \(\mathcal{C} X\).

     

Now let \(S = \{a_1 < a_2 < \dots < a_n\} \subset {\mathbb{R}}\) for some non-negative integer \(n\). The definition of a homomorphism between \(S\)-graphs is completely analogous to definition 28, homomorphisms between \(S\)-skeletons for \(\overline{{\mathbb{R}}}\)-spaces. More precisely the full subcategory of the category of \(S\)-skeletons for \(\overline{{\mathbb{R}}}\)-spaces with all \(V_i\) and \(E_i\) discrete is isomorphic to the category of \(S\)-graphs. For convenience we spell out the details nevertheless.

  • Definition. Let \(G\) and \(G'\) be two \(S\)-graphs and suppose we have the following data:

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

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

    This data describes a homomorphism of \(S\)-graphs \(\varphi \colon G \rightarrow G'\), 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 \(S\)-graphs is defined by composing the individual maps \(\varphi^v_i\) and \(\varphi^e_i\).

The functor \(|\_|\) defined on the category of \(S\)-skeletons for \(\overline{{\mathbb{R}}}\)-spaces from the previous section restricts to the category of \(S\)-graphs, if we identify the category of \(S\)-graphs with the full subcategory of the category of \(S\)-skeletons for \(\overline{{\mathbb{R}}}\)-spaces with all \(V_i\) and \(E_i\) discrete.

  • Lemma. The induced functor \(|\_|\) on \(S\)-graphs as described above is full and faithful.