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