A simplicial map (also called simplicial mapping) is a function between two simplicial complexes, with the property that the images of the vertices of a simplex always span a simplex.
[1] Simplicial maps can be used to approximate continuous functions between topological spaces that can be triangulated; this is formalized by the simplicial approximation theorem.
A simplicial map is defined in slightly different ways in different contexts.
Let K and L be two abstract simplicial complexes (ASC).
[2]: 14, Def.1.5.2 As an example, let K be the ASC containing the sets {1,2},{2,3},{3,1} and their subsets, and let L be the ASC containing the set {4,5,6} and its subsets.
is not bijective, it may map k-dimensional simplices in K to l-dimensional simplices in L, for any l ≤ k. In the above example, f maps the one-dimensional simplex {1,2} to the zero-dimensional simplex {4}.
is called a simplicial isomorphism.
Isomorphic simplicial complexes are essentially "the same", up to a renaming of the vertices.
The existence of an isomorphism between L and K is usually denoted by
[2]: 14 The function f defined above is not an isomorphism since it is not bijective.
If we modify the definition to f(1)=4, f(2)=5, f(3)=6, then f is bijective but it is still not an isomorphism, since
, which is not a simplex in K. If we modify L by removing {4,5,6}, that is, L is the ASC containing only the sets {4,5},{5,6},{6,4} and their subsets, then f is an isomorphism.
Let K and L be two geometric simplicial complexes (GSC).
A simplicial map of K into L is a function
Note that this implies that vertices of K are mapped to vertices of L. [1] Equivalently, one can define a simplicial map as a function from the underlying space of K (the union of simplices in K) to the underlying space of L,
[3]: 16 [4]: 3 Every simplicial map is continuous.
Simplicial maps are determined by their effects on vertices.
In particular, there are a finite number of simplicial maps between two given finite simplicial complexes.
A simplicial map between two ASCs induces a simplicial map between their geometric realizations (their underlying polyhedra) using barycentric coordinates.
be its support (the unique simplex containing x in its interior), and denote the vertices of
has a unique representation as a convex combination of the vertices,
This |f| is a simplicial map of |K| into |L|; it is a continuous function.
be a continuous map between the underlying polyhedra of simplicial complexes and let us write
, is called a simplicial approximation to
See simplicial approximation theorem for more details.
Every simplicial map is PL, but the opposite is not true.
be a non-linear function that maps the leftmost half of |K| linearly into the leftmost half of |L|, and maps the rightmost half of |K| linearly into the rightmost half of |L|.
Then f is PL, since it is a simplicial map between a subdivision of |K| into two triangles and a subdivision of |L| into two triangles.
This notion is an adaptation of the general notion of a piecewise-linear function to simplicial complexes.
A PL homeomorphism between two polyhedra |K| and |L| is a PL mapping such that the simplicial mapping between the subdivisions,