### 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.