# Simplicial map

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

Let be a continuous map between the underlying polyhedra of simplicial complexes and let us write for the star of a vertex. A simplicial map such that , is called a **simplicial approximation** to .

A simplicial approximation is homotopic to the map it approximates.

## References

- Munkres, James R.:
*Elements of Algebraic Topology*, Westview Press, 1995. ISBN 978-0-201-62728-2.