Simplicial map
Encyclopedia
In the mathematical discipline of simplicial homology
theory, a simplicial map is a map
between simplicial complex
es with the property that the images of the vertices of a simplex
always span a simplex. 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 maps which are bijective
are called simplicial isomorphisms
.
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.
Simplicial homology
In mathematics, in the area of algebraic topology, simplicial homology is a theory with a finitary definition, and is probably the most tangible variant of homology theory....
theory, a simplicial map is a map
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
between simplicial complex
Simplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
es with the property that the images of the vertices of a simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
always span a simplex. 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 maps which are bijective
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
are called simplicial isomorphisms
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
.
Simplicial approximation
Let be a continuous map between the underlying polyhedra of simplicial complexes and let us write for the starSimplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
of a vertex. A simplicial map
such that , is called a simplicial approximation to .A simplicial approximation is homotopic
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
to the map it approximates.