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} $.

Comments

For references see Simplicial set.