Geometric realization of a simplicial set ($|X|$)

The geometric realization of a simplicial set glues one topological simplex for each simplicial simplex, using the face and degeneracy maps as the gluing relations.

After sufficiently subdividing the domain, every continuous map between geometric realizations of simplicial complexes is homotopic to a simplicial map. Relative and based forms preserve a specified subcomplex or basepoint.