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.<ref name=":1">{{cite book |last=Munkres |first=James R. |title=Elements of Algebraic Topology |publisher=Westview Press |year=1995 |isbn=978-0-201-62728-2 |authorlink=James Munkres}}</ref> 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,'' <math>f: V(K)\to V(L)</math>, that maps every simplex in K to a simplex in L. That is, for any <math>\sigma\in K</math>, <math>f(\sigma)\in L</math>.''<ref name=":0">{{Cite Matousek 2007}}, Section 4.3</ref>''{{Rp|page=14|location=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. 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 <math>f</math> 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 <math>f</math> is bijective, and its inverse <math>f^{-1}</math> is a simplicial map of L into K, then <math>f</math> 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 <math>K\cong L</math>.''<ref name=":0" />''{{Rp|page=14|location=}} 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 <math>f^{-1}</math> is not simplicial: <math>f^{-1}(\{4,5,6\})= \{1,2,3\}</math>, 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 <math>f: K\to L</math> such that the images of the vertices of a simplex in K span a simplex in L. That is, for any simplex <math>\sigma\in K</math>, <math>\operatorname{conv}(f(V(\sigma)))\in L</math>. Note that this implies that vertices of K are mapped to vertices of L. <ref name=":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, <math>f: |K|\to |L|</math>, that maps every simplex in K ''linearly'' to a simplex in L. That is, for any simplex <math>\sigma\in K</math>, <math>f(\sigma)\in L</math>, and in addition, <math>f\vert_{\sigma}</math> (the restriction of <math>f</math> to <math>\sigma</math>) is a linear function.<ref>{{Cite book |last=Colin P. Rourke and Brian J. Sanderson |url=https://link.springer.com/book/10.1007/978-3-642-81735-9 |title=Introduction to Piecewise-Linear Topology |publisher=Springer-Verlag |year=1982 |location=New York |language=en |doi=10.1007/978-3-642-81735-9|isbn=978-3-540-11102-3 }}</ref>{{Rp|page=16}}<ref>{{Citation |last=Bryant |first=John L. |title=Chapter 5 - Piecewise Linear Topology |date=2001-01-01 |url=https://www.sciencedirect.com/science/article/pii/B9780444824325500068 |work=Handbook of Geometric Topology |pages=219–259 |editor-last=Daverman |editor-first=R. J. |place=Amsterdam |publisher=North-Holland |language=en |isbn=978-0-444-82432-5 |access-date=2022-11-15 |editor2-last=Sher |editor2-first=R. B.}}</ref>{{Rp|page=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. This can be defined precisely.''<ref name=":0" />''{{Rp|page=15|location=Def.1.5.3}} Let K, L be two ASCs, and let <math>f: V(K)\to V(L)</math> be a simplicial map. The '''affine extension''' of <math>f</math> is a mapping <math>|f|: |K|\to |L|</math> defined as follows. For any point <math>x\in |K|</math>, let <math>\sigma</math> be its support (the unique simplex containing ''x'' in its interior), and denote the vertices of <math>\sigma</math> by <math>v_0,\ldots,v_k</math>. The point <math>x</math> has a unique representation as a convex combination of the vertices, <math>x = \sum_{i=0}^k a_i v_i</math> with <math>a_i \geq 0 </math> and <math>\sum_{i=0}^k a_i = 1</math> (the <math>a_i</math> are the barycentric coordinates of <math>x</math>). We define <math>|f|(x) := \sum_{i=0}^k a_i f(v_i)</math>. 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''|.''<ref name=":0" />''{{Rp|page=15|location=Prop.1.5.4}}
==Simplicial approximation== Let <math>f\colon |K| \to |L|</math> be a continuous map between the underlying polyhedra of simplicial complexes and let us write <math>\text{st}(v)</math> for the star of a vertex. A simplicial map <math>f_\triangle\colon K \to L</math> such that <math>f(\text{st}(v)) \subseteq \text{st}(f_\triangle (v))</math>, is called a '''simplicial approximation''' to <math>f</math>.
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 <math>f: |K|\to |L|</math> is called '''piecewise-linear''' '''(PL)''' if there exist a subdivision ''K''<nowiki/>' of ''K'', and a subdivision ''L''<nowiki/>' of ''L'', such that <math>f: |K'|\to |L'|</math> 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 <math>f: |K|\to |L|</math> 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, <math>f: |K'|\to |L'|</math>, is a homeomorphism.
==References== {{Reflist}} Category:Algebraic topology Category:Simplicial homology Category:Simplicial sets