Simplicial map

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In the mathematical discipline of simplicial homology theory, a simplicial map is a map between simplicial complexes with the property that the images of the vertices of a simplex always span a simplex. Note that this implies that vertices have vertices for images.

Simplicial maps are thus determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.

Simplicial maps induce continuous maps between the underlying polyhedra of the simplicial complexes: one simply extends linearly using barycentric coordinates.

Simplicial maps which are bijective are called simplicial isomorphisms.

Simplicial approximation[edit]

Let f: |K| \rightarrow |L| be a continuous map between the underlying polyhedra of simplicial complexes and let us write \text{st}(v) for the star of a vertex. A simplicial map f_\triangle :K \rightarrow L such that f(\text{st}(v)) \subseteq \text{st}(f_\triangle (v)), is called a simplicial approximation to f.

A simplicial approximation is homotopic to the map it approximates.

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