Simplicial map
Introduction 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. 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 isomorphism''' is a bijective simplicial map such that both it and its inverse are simplicial.
Definitions A simplicial map is defined in slightly different ways in different contexts.=== Abstract simplicial complexes ===Let K and L be two abstract simplicial complexes (ASC). A '''simplicial map''' '''of K into L''' is a function from the vertices of ''K'' to the vertices of ''L,'' , that maps every simplex in K to a simplex in L. That is, for any , .'''' As an example, let K be 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. Define a mapping ''f'' by:  ''f''(1)=''f''(2)=4, ''f''(3)=5.Then ''f'' is a simplicial mapping, since ''f''({1,2})={4} which is a simplex in L, ''f''({2,3})=f({3,1})={4,5} which is also a simplex in L, etc.If  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}.If  is bijective, and its inverse  is a simplicial map of L into K, then  is called a '''simplicial isomorphism'''. Isomorphic simplicial complexes are essentially "the same", up ro a renaming of the vertices. The existence of an isomorphism between L and K is usually denoted by .'''' 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  is not simplicial: , 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.=== Geometric simplicial complexes ===Let K and L be two geometric simplicial complexes (GSC). A '''simplicial map''' '''of K into L''' is a function  such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex ,  . Note that this implies that vertices of K are mapped to vertices of L. 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,  , that maps every simplex in K ''linearly'' to a simplex in L. That is, for any simplex , , and in addition,  (the restriction of  to ) is a linear function. 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. This can be defined precisely.'''' Let K, L be to ASCs, and let  be a simplicial map. The '''affine extension''' of  is a mapping  defined as follows. For any point , let  be its support (the unique simplex containing ''x'' in its interior), and denote the vertices of  by . The point  has a unique representation as a convex combination of the vertices,  with  and  (the  are the barycentric coordinates of ). We define . This |''f''| is a simplicial map of |K| into |L|; it is a continuous function. If ''f'' is injective, then |''f''| is injective; if ''f'' is an isomorphism between ''K'' and ''L'', then |''f''| is a homeomorphism between |''K''| and |''L''|.''''
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. See simplicial approximation theorem for more details.
Piecewise-linear maps Let K and L be two GSCs. A function  is called '''piecewise-linear''' '''(PL)''' if there exist a subdivision ''K''' of ''K'', and a subdivision ''L''' of ''L'', such that  is a simplicial map of K' into L'. Every simplicial map is PL, but the opposite is not true. For example, suppose |K| and |L| are two triangles, and let  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 rightmostt 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, , is a homeomorphism.
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