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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.

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

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

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).

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

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

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|Simplicial set]].