Definition A simplicial map ϕ: K → L between simplicial complexes K and L is a function ϕ: Vert K → Vert L from the vertex set of K to that of L such that ϕ(v₀), ϕ(v₁), . . . , ϕ(v_q) span a simplex belonging to L whenever v₀, v₁, . . . , v_q span a simplex of K.

Note that a simplicial map ϕ: K → L between simplicial complexes K and L can be regarded as a function from K to L: this function sends a simplex σ of K with vertices v₀, v₁, . . . , v_q to the simplex ϕ(σ) of L spanned by the vertices ϕ(v₀), ϕ(v₁), . . . , ϕ(v_q).

A simplicial map ϕ: K → L also induces in a natural fashion a continuous map ϕ: |K| → |L| between the polyhedra of K and L, where

whenever 0 ≤ t_j ≤ 1 for j = 0, 1, . . . , q,

simplex of K. The continuity of this map follows immediately from a straight forward application of Lemma 5.2. Note that the interior of a simplex σ of K is mapped into the interior of the simplex ϕ(σ) of L.