A simplicial map sends vertices to vertices and sends the vertex set of every simplex to the vertex set of a simplex. It induces a continuous map between geometric realizations and a chain map between simplicial chain complexes.
Simplicial maps are contiguous when lies in one simplex of for every simplex of . Contiguous maps have homotopic geometric realizations.
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In the context of algebraic topology and category theory, a **simplicial map** is a function between simplicial sets that preserves the structure of simplicial complexes. To understand this more formally, let's break it down: ### Simplicial Sets and Simplicial Complexes 1. **Simplicial Complex**: A simplicial complex is a set composed of "simplices" (generalized triangles) that satisfy certain properties.