Definition. One-dimensional realization of finite incidence [fdip-004M]

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Let \(G=(V,E,s,h)\) be a finite incidence presentation. Its one-dimensional realization is the quotient space \[|G|=\bigl (V\sqcup (E\times [0,1])\bigr )/{\sim },\] where \(V,E\) are discrete, the intervals have their usual topology, and the equivalence relation is generated by \[(e,0)\sim s(e),\qquad (e,1)\sim h(e).\] Use the quotient topology. Thus each vertex is a point and each edge supplies an open interval attached at its endpoints, the standard one-dimensional CW construction of [hatcher2002algebraic, Section 1.A, p. 83]. Loops have both endpoints attached at one vertex; parallel edges remain separate intervals. Direction chooses the parametrization from \(0\) to \(1\), not a causal ordering.

Declare a morphism \(f:G\to G'\) to be a pair of maps \(f_V:V\to V'\) and \(f_E:E\to E'\) with \[f_Vs=s'f_E,\qquad f_Vh=h'f_E.\] It induces \[\begin {aligned} |f|:|G|&\longrightarrow |G'|,\\ [v]&\longmapsto [f_V(v)],\\ [e,t]&\longmapsto [f_E(e),t]. \end {aligned}\] These formulas respect the endpoint identifications. They are continuous before quotienting, so the quotient topology makes \(|f|\) continuous. Identity and composition follow from the formulas. This proves functoriality for these declared incidence maps; arbitrary edge contractions have not been included among them. The map construction is a direct deduction from the quoted quotient model.