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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 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 complexs

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, $f:V(K)\to V(L)$ , that maps every simplex in K to a simplex in L. That is, for any $\sigma \in K$ , $f(\sigma )\in L$ .^[2]^{: 14, Def.1.5.2} 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})={4,5} which is also a simplex in L, etc.

If $f$ is bijective, and its inverse $f^{-1}$ is a simplicial map of L into K, then $f$ 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 $K\cong L$ .^[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 $f^{-1}$ is not simplicial: $f^{-1}(\{4,5,6\})=\{1,2,3\}$ , 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.

Geometric simplicial complexes

Let K and L be two geometric simplicial complexes. A simplicial map of K into L is a function from the underlying space of K (the union of simplices in K) to the underlying space of L, $f:|K|\to |L|$ , that maps every simplex in K linearly to a simplex in L. That is, for any $\sigma \in K$ , $f(\sigma )\in L$ .^[2]^{: 14, Def.1.5.2}

^[3]^: 5

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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 approximation

Let $f\colon |K|\to |L|$ be a continuous map between the underlying polyhedra of simplicial complexes and let us write ${\text{st}}(v)$ for the star of a vertex. A simplicial map $f_{\triangle }\colon K\to L$ such that $f({\text{st}}(v))\subseteq {\text{st}}(f_{\triangle }(v))$ , is called a simplicial approximation to $f$ .

A simplicial approximation is homotopic to the map it approximates.

References

^ Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2.
^ ^a ^b ^c Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3
^ Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15
^ Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9.

[1] Munkres, James R. (1995). Elements of Algebraic Topology. Westview Press. ISBN 978-0-201-62728-2.

[:0-2] Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3

[3] Bryant, John L. (2001-01-01), Daverman, R. J.; Sher, R. B. (eds.), "Chapter 5 - Piecewise Linear Topology", Handbook of Geometric Topology, Amsterdam: North-Holland, pp. 219–259, ISBN 978-0-444-82432-5, retrieved 2022-11-15

[4] Colin P. Rourke and Brian J. Sanderson (1982). Introduction to Piecewise-Linear Topology. New York: Springer-Verlag. doi:10.1007/978-3-642-81735-9.

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-A '''simplicial map''' is a [[Function (mathematics)|function]] between two [[simplicial complexes]], with the property that the images of the vertices of a [[simplex]] always span a simplex.<ref>{{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 [[Triangulation (topology)|triangulated]]; this is formalized by the [[simplicial approximation theorem]].
+A '''simplicial map''' (also called '''simplicial mapping''') is a [[Function (mathematics)|function]] between two [[simplicial complexes]], with the property that the images of the vertices of a [[simplex]] always span a simplex.<ref>{{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 [[Triangulation (topology)|triangulated]]; this is formalized by the [[simplicial approximation theorem]].
+A '''simplicial isomorphism''' is a [[bijection|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 complexs ===
+Let K and L be two [[Abstract simplicial complex|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 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})={4,5} which is also a simplex in L, etc.
+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 ro 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 complex]]<nowiki/>es. A '''simplicial map''' '''of K into L''' is 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 <math>\sigma\in K</math>, <math>f(\sigma)\in L</math>.''<ref name=":0" />''{{Rp|page=14|location=Def.1.5.2}}
+<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=5}}
+<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}}</ref>
 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 coordinate system (mathematics)|barycentric coordinates]].
-Simplicial maps which are [[bijection|bijective]] are called ''simplicial [[isomorphism]]s''.
 ==Simplicial approximation==