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The [[Simplex|simplex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871701.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871702.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871703.png" /> with vertices at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871705.png" /> (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871706.png" /> stands in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871707.png" />-th place), i.e.
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871708.png" /></td> </tr></table>
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For any topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s0871709.png" />, the continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717010.png" /> are the singular simplices of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717011.png" /> (see [[Singular homology|Singular homology]]).
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The [[Simplex|simplex]]  $  \Delta  ^ {n} $
 +
of dimension  $  n $
 +
in the space $  \mathbf R  ^ {n+} 1 $
 +
with vertices at the points  $  e _ {i} = ( 0 \dots 1 \dots 0) $,  
 +
$  i = 0 \dots n $(
 +
the  $  1 $
 +
stands in the $  i $-
 +
th place), i.e.
  
The [[Simplicial complex|simplicial complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717012.png" /> whose vertices are the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717014.png" />, while the simplices are arbitrary non-empty subsets of vertices. The geometric realization of this simplicial complex coincides with the standard simplex in the sense of 1).
+
$$
 +
\Delta  ^ {n}  = \{ {( t _ {0} \dots t _ {n+} 1 ) } : {0 \leq  t _ {i} \leq  1, \sum t _ {i} = 1 } \}
 +
\subset  \mathbf R  ^ {n+} 1 .
 +
$$
  
The [[Simplicial set|simplicial set]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717015.png" />, obtained by applying the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717016.png" /> to the simplicial scheme in 2), which is a contra-variant functor on the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717017.png" /> (see [[Simplicial object in a category|Simplicial object in a category]]), for which
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For any topological space  $  X $,  
 +
the continuous mappings  $  \sigma : \Delta  ^ {n} \rightarrow X $
 +
are the singular simplices of  $  X $(
 +
see [[Singular homology|Singular homology]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717018.png" /></td> </tr></table>
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The [[Simplicial complex|simplicial complex]]  $  \Delta  ^ {n} $
 +
whose vertices are the points  $  l _ {i} $,
 +
$  0 \leq  i \leq  n $,
 +
while the simplices are arbitrary non-empty subsets of vertices. The geometric realization of this simplicial complex coincides with the standard simplex in the sense of 1).
  
Thus, non-decreasing sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717019.png" /> of numbers from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717020.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717021.png" />-dimensional simplices of the simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717022.png" />, while the face operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717023.png" /> and the degeneracy operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717024.png" /> of this simplicial set are defined by the formulas
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The [[Simplicial set|simplicial set]]  $  \Delta  ^ {n} $,  
 +
obtained by applying the functor  $  O  ^ {+} $
 +
to the simplicial scheme in 2), which is a contra-variant functor on the category  $  \Delta $(
 +
see [[Simplicial object in a category|Simplicial object in a category]]), for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717025.png" /></td> </tr></table>
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$$
 +
\Delta  ^ {n} ([ m])  = \Delta ([ m], [ n]),\ \
 +
\Delta  ^ {n} ( \lambda )( \mu )  = \mu \lambda .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717026.png" /></td> </tr></table>
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Thus, non-decreasing sequences  $  ( a _ {0} \dots a _ {m} ) $
 +
of numbers from  $  [ n] $
 +
are  $  m $-
 +
dimensional simplices of the simplicial set  $  \Delta  ^ {n} $,
 +
while the face operators  $  d _ {i} $
 +
and the degeneracy operators  $  s _ {i} $
 +
of this simplicial set are defined by the formulas
  
where the sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717027.png" /> signifies that the symbol beneath it is deleted. The simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717028.png" /> is also called a simplicial segment. The simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717029.png" /> (the unique non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717030.png" />-dimensional simplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717031.png" />) is called the fundamental simplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717032.png" />. The smallest simplicial subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717033.png" /> containing all simplices of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717035.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717036.png" /> and is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717038.png" />-th standard horn.
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$$
 +
d _ {i} ( a _ {0} \dots a _ {m} )  = ( a _ {0} \dots a _ {i-} 1 , \widehat{a}  _ {i} , a _ {i+} 1 \dots a _ {m} ),
 +
$$
  
For any simplicial set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717039.png" /> and an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717040.png" />-dimensional simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717042.png" />, there is a unique simplicial mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717044.png" />. This mapping is said to be characteristic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717045.png" />.
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$$
 +
s _ {i} ( a _ {0} \dots a _ {m} )  = ( a _ {0} \dots a _ {i} , a _ {i} , a _ {i+} 1 \dots a _ {m} ),
 +
$$
  
The fundamental simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717046.png" /> of a simplicial set as in 3), which in this instance is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087170/s08717047.png" />.
+
where the sign  $  \widehat{ {}}  $
 +
signifies that the symbol beneath it is deleted. The simplicial set  $  \Delta  ^ {1} $
 +
is also called a simplicial segment. The simplex $  \iota _ {n} = ( 0, 1 \dots n) $(
 +
the unique non-degenerate  $  n $-
 +
dimensional simplex of  $  \Delta  ^ {n} $)
 +
is called the fundamental simplex of  $  \Delta  ^ {n} $.  
 +
The smallest simplicial subset of  $  \Delta  ^ {n+} 1 $
 +
containing all simplices of the form  $  d _ {i} \iota _ {n+} 1 $
 +
with  $  i \neq k $
 +
is denoted by $  \Delta _ {k}  ^ {n} $
 +
and is called the  $  k $-
 +
th standard horn.
  
 +
For any simplicial set  $  K $
 +
and an arbitrary  $  n $-
 +
dimensional simplex  $  \sigma $
 +
of  $  K $,
 +
there is a unique simplicial mapping  $  \chi _  \sigma  :  \Delta  ^ {n} \rightarrow K $
 +
for which  $  \chi ( \iota _ {n} ) = \sigma $.
 +
This mapping is said to be characteristic for  $  \sigma $.
  
 +
The fundamental simplex  $  \iota _ {n} $
 +
of a simplicial set as in 3), which in this instance is denoted by  $  \Delta _ {n} $.
  
 
====Comments====
 
====Comments====
 
For references see [[Simplicial set|Simplicial set]].
 
For references see [[Simplicial set|Simplicial set]].

Revision as of 08:22, 6 June 2020


The simplex $ \Delta ^ {n} $ of dimension $ n $ in the space $ \mathbf R ^ {n+} 1 $ with vertices at the points $ e _ {i} = ( 0 \dots 1 \dots 0) $, $ i = 0 \dots n $( the $ 1 $ stands in the $ i $- th place), i.e.

$$ \Delta ^ {n} = \{ {( t _ {0} \dots t _ {n+} 1 ) } : {0 \leq t _ {i} \leq 1, \sum t _ {i} = 1 } \} \subset \mathbf R ^ {n+} 1 . $$

For any topological space $ X $, the continuous mappings $ \sigma : \Delta ^ {n} \rightarrow X $ are the singular simplices of $ X $( see Singular homology).

The simplicial complex $ \Delta ^ {n} $ whose vertices are the points $ l _ {i} $, $ 0 \leq i \leq n $, while the simplices are arbitrary non-empty subsets of vertices. The geometric realization of this simplicial complex coincides with the standard simplex in the sense of 1).

The simplicial set $ \Delta ^ {n} $, obtained by applying the functor $ O ^ {+} $ to the simplicial scheme in 2), which is a contra-variant functor on the category $ \Delta $( see Simplicial object in a category), for which

$$ \Delta ^ {n} ([ m]) = \Delta ([ m], [ n]),\ \ \Delta ^ {n} ( \lambda )( \mu ) = \mu \lambda . $$

Thus, non-decreasing sequences $ ( a _ {0} \dots a _ {m} ) $ of numbers from $ [ n] $ are $ m $- dimensional simplices of the simplicial set $ \Delta ^ {n} $, while the face operators $ d _ {i} $ and the degeneracy operators $ s _ {i} $ of this simplicial set are defined by the formulas

$$ d _ {i} ( a _ {0} \dots a _ {m} ) = ( a _ {0} \dots a _ {i-} 1 , \widehat{a} _ {i} , a _ {i+} 1 \dots a _ {m} ), $$

$$ s _ {i} ( a _ {0} \dots a _ {m} ) = ( a _ {0} \dots a _ {i} , a _ {i} , a _ {i+} 1 \dots a _ {m} ), $$

where the sign $ \widehat{ {}} $ signifies that the symbol beneath it is deleted. The simplicial set $ \Delta ^ {1} $ is also called a simplicial segment. The simplex $ \iota _ {n} = ( 0, 1 \dots n) $( the unique non-degenerate $ n $- dimensional simplex of $ \Delta ^ {n} $) is called the fundamental simplex of $ \Delta ^ {n} $. The smallest simplicial subset of $ \Delta ^ {n+} 1 $ containing all simplices of the form $ d _ {i} \iota _ {n+} 1 $ with $ i \neq k $ is denoted by $ \Delta _ {k} ^ {n} $ and is called the $ k $- th standard horn.

For any simplicial set $ K $ and an arbitrary $ n $- dimensional simplex $ \sigma $ of $ K $, there is a unique simplicial mapping $ \chi _ \sigma : \Delta ^ {n} \rightarrow K $ for which $ \chi ( \iota _ {n} ) = \sigma $. This mapping is said to be characteristic for $ \sigma $.

The fundamental simplex $ \iota _ {n} $ of a simplicial set as in 3), which in this instance is denoted by $ \Delta _ {n} $.

Comments

For references see Simplicial set.

How to Cite This Entry:
Standard simplex. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Standard_simplex&oldid=48801
This article was adapted from an original article by S.N. MalyginM.M. Postnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article