# Standard simplex

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 $\Delta _ {n}$.