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, f : V ( K ) → V ( L ) {\displaystyle f:V(K)\to V(L)} , that maps every simplex in K to a simplex in L. That is, for any σ ∈ K {\displaystyle \sigma \in K} , f ( σ ) ∈ L {\displaystyle f(\sigma )\in L} . 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. 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 f {\displaystyle f} 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 f {\displaystyle f} is bijective, and its inverse f − 1 {\displaystyle f^{-1}} is a simplicial map of L into K, then f {\displaystyle f} 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 K ≅ L {\displaystyle K\cong L} . 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 f − 1 {\displaystyle f^{-1}} is not simplicial: f − 1 ( { 4 , 5 , 6 } ) = { 1 , 2 , 3 } {\displaystyle f^{-1}(\{4,5,6\})=\{1,2,3\}} , 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.
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